Absolute Continuity and Singular Measures

Prove the following variant of Egorov’s theorem that applies to sets of infinite measure. Let \((X,\mathcal {M},\mu )\) be a measure space and let \(g\in L^1(\mu )\). Suppose \(f_n:X\to \mathbb {R}\) are measurable with \(|f_n|\le g\) and \(f_n\to f\) pointwise. Then for every \(\epsilon >0\) there is a set \(E\subset X\) of measure \(<\epsilon \) so that \(f_n\to f\) uniformly on \(X\setminus E\).

Let \(\mu \) be a Borel measure on \(\mathbb {R}\) (i.e. a measure defined on the Borel \(\sigma \)-algebra). Suppose it satisfies:

1. \[ \mu ([0,1])=1. \]
2. \(\mu (A+t)=\mu (A)\) for every Borel set \(A\) and every \(t\in \mathbb {R}\), i.e. \(\mu \) is translation invariant.

Show that \(\mu \) equals the Lebesgue measure.

Construct a compact subset in \(\mathbb {R}^2\) with empty interior and with positive Lebesgue measure.

Let \(X\) be a set and \(S\subset \mathcal {P}(X)\) a collection of subsets of \(X\) containing the empty set and the whole space. Let \(\psi :S\to [0,\infty ]\) be a function such that \(\psi (\emptyset )=0\). Define \(\psi ^*:\mathcal {P}(X)\to [0,\infty ]\) as \[ \psi ^*(A)=\inf \left \{\sum _{i=1}^{\infty }\psi (S_i): A\subseteq \bigcup _{i=1}^{\infty }S_i\right \}. \]

Define what it means to be an outer measure and show that \(\psi ^*\) is an outer measure.

Let

Show that \(A\) has Lebesgue measure 0 but

Prove the following generalization of the Dominated Convergence Theorem. Let \((X,\mathcal {M},\mu )\) be a measure space. Suppose \(f_n,g_n,f,g:X\to [0,\infty )\) are measurable and have finite integrals, \(f_n\to f\), \(g_n\to g\) (pointwise convergence), \(f_n\le g_n\), \(\int g_n\to \int g<\infty \). Then \(\int f_n\to \int f\).

Let \(\mu \) and \(\nu \) be two finite positive measures on a measurable space \((X,\mathcal {M})\). Suppose for every \(\epsilon >0\), there exists \(E\in \mathcal {M}\) such that \(\mu (E)<\epsilon \) and \(\nu (X\setminus E)<\epsilon \). Show that \(\mu \perp \nu \) (\(\mu \) and \(\nu \) are mutually singular).

Let \((X,\mathcal {M},\mu )\) be a measure space with \(\mu (X)<\infty \). Show that if \(1\le q< p\le \infty \) then \(L^q(X)\supseteq L^p(X)\). Moreover, if \(X\) contains an infinite family of pairwise disjoint subsets of positive measure, then \(L^q(X)\supsetneq L^p(X)\).

Let \(X\) be a set and \(\mu ^*\) be an outer measure on \(X\). A set \(A\subset X\) is called \(\mu ^*\)-measurable if for every \(S\subset X\) we have

Show that \(\mu ^*\) is countably additive on the \(\mu ^*\) measurable sets.

We say \(f_1,\ldots \) is \(\lambda \)-Cauchy if for every \(\epsilon >0\) there exists \(N\) so that for \(i,j>N\) we have

Show that if \(f_1,\ldots \in L^1(\lambda )\) is \(\lambda \)-Cauchy then there exists \(g\in L^1(\lambda )\) so that \(f_1,f_2,\ldots \) converges in measure to \(g\).

Let \((X,\mathcal {M})\) be a measurable space and \(\mu \) and \(\nu \) be two finite measures on \((X,\mathcal {M})\). Show that the following are equivalent

• \(\nu \ll \mu \)
• Whenever \(A_1,\ldots \in \mathcal {M}\) satisfy \(\lim _{n\to \infty }\mu (A_n)=0\) we have \(\lim _{n\to \infty }\nu (A_n)=0\).

Let \(X=[0,1]\) and let \(\mu \) be a finite Borel measure on \(X\). Prove that \(\mu \) is regular, i.e. that for any Borel set \(A\subseteq X\) the following holds:

1. \(\mu (A)=\sup \{\mu (K)\mid K\subseteq A,\ K\text { compact}\}\), and
2. \(\mu (A)=\inf \{\mu (U)\mid U\supseteq A,\ U\text { open}\}\).

Note: The same statement holds for any compact metrizable space \(X\).

Let \((X,\mathcal {B},\mu )\) be a finite measure space and \(f:X\to \mathbb {R}\) be measurable.

1. Show that if \(g_n(x)=(f(x))^n\) and \(\int g_n\,d\mu \) is uniformly bounded for all \(n\) then \(|f(x)|\le 1\) (\(\mu \)-a.e.).
2. Show that \(\int _X g_n\,d\mu \) is independent of \(n\) iff there is \(A\in \mathcal {B}\) so that \(f=\chi _A\) (\(\mu \) a.e.).

Let \(X\) be a set and \(\mathcal {A}\) be an algebra on \(X\). Let \(\rho _0:\mathcal {A}\to [0,\infty ]\) be a pre-measure on \(\mathcal {A}\). Recall that \[ \rho ^*(S)=\inf \left \{\sum _{i=1}^{\infty }\rho _0(A_i): A_i\in \mathcal {A}\text { for all }i\text { and }S\subset \bigcup _{i=1}^{\infty }A_i\right \} \]

is an outer measure. Define what it means that \(\rho _0\) is a pre-measure and show that if \(A\in \mathcal {A}\) then \(\rho ^*(A)=\rho _0(A)\).

Let \((X,\mathcal {M},\mu )\) and \((Y,\mathcal {N},\nu )\) be two measure spaces and \(A\) be measurable with respect to \(\mathcal {M}\times \mathcal {N}\), the \(\sigma \)-algebra generated by measurable rectangles. Show that if \(A_1\subset A_2\subset \cdots \) satisfy that \[ (\mu \otimes \nu )(A_i)=\int _X \nu (\{y:(x,y)\in A_i\})\,d\mu (x) \]

for all \(i\) then

Let \((X,\mathcal {M},\mu )\) be a measure space. Give an example of \(f_1,\ldots :X\to \mathbb {C}\), measurable, which converge in measure to the zero function but do not converge \(\mu \)-a.e. Show that there exists a subsequence \(n_1,n_2,\ldots \) so that \(f_{n_1},\ldots \) converges almost everywhere to the zero function.

Let \((X, \mathcal {M}, \mu )\) be a measure space and \(f : X \to Y\) be a function.

1. Show that \(\mathcal {N} = \{f^{-1}A : A \in \mathcal {M}\}\) is a \(\sigma \)-algebra.
2. Let \(\nu (B) = \mu (f(B))\) for \(B \in \mathcal {N}\). Show that \(\nu \) is a measure.
3. Show that even if \((X, \mathcal {M}, \mu )\) is complete we do not necessarily have that \((Y, \mathcal {N}, \nu )\) is complete.

Let \((X, \mathcal {T})\) be a topological space and \(\mathcal {B}\) be its Borel \(\sigma \)-algebra. Show that if \((X, \mathcal {B}, \mu )\) is a \(\sigma \)-finite measure space that is outer regular on finite measure, measurable subsets, then it is automatically outer regular on all measurable subsets. Recall that \(\mu \) is called outer regular on a collection of subsets \(\mathcal {S}\) if for every \(A \in \mathcal {S}\) and \(\epsilon > 0\) there is an open set \(U\) so that \(A \subset U\) and \(\mu (U \setminus A) < \epsilon \).

Prove the Borel-Cantelli lemma: Let \((X, \mathcal {M}, \mu )\) be a measure space and let \(A_{i}\) be measurable sets such that \(\sum _{i=1}^{\infty }\mu (A_{i}) < \infty \). Then the set

of points that belong to infinitely many \(A_{i}\) has measure 0.

Let \(X\) be a compact metric space, \(\mathcal {M}\) the Borel \(\sigma \)-algebra on \(X\), and \(\mu \) a finite measure on \(\mathcal {M}\) (meaning that \(\mu (X) < \infty \); we call such a measure a finite Borel measure on \(X\)). Prove that \(\mu \) is regular, i.e. for every Borel set \(E \subset X\):

• \(\mu (E) = \sup \{\mu (K) \mid K \subseteq E \text { is compact}\}\), and
• \(\mu (E) = \inf \{\mu (U) \mid U \supseteq E \text { is open}\}\).

Hint: Show that the set of \(E\)’s that satisfy both bullets is a \(\sigma \)-algebra that contains compact sets.

1. Let \((X, \mathcal {M})\) be a measurable space, \(\mu , \nu \) positive measures on \((X, \mathcal {M})\). Suppose \(\nu (X) < \infty \). Prove that \(\nu \) \(\ll \mu \) if and only if for every \(\epsilon > 0\) there exists \(\delta > 0\) such that \(\mu (E) < \delta \) implies \(\nu (E) < \epsilon \).
2. For \(\mu \) the Lebesgue measure on \(\mathbb {R}\) find a positive (but infinite) measure \(\nu \) such that \(\nu \) \(\ll \mu \) but the \(\epsilon \)-\(\delta \) statement in (a) fails.

Let \((X, \mathcal {M}, \mu )\) be a measure space with \(\mu (X) < \infty \). Let \(f : X \to X\) be a measure preserving bijection, in the sense that \(\mu (A) = \mu (f(A))\) for every \(A \in \mathcal {M}\). Prove that for every \(E \in \mathcal {M}\)

has measure 0.

Let \((X, \mathcal {M}, \mu )\) be a measure space with \(\mu (X) = 1\) and let \(E_{1}, \ldots , E_{n}\) be measurable subsets of \(X\). Suppose that every \(x \in X\) belongs to at least \(k\) of the sets \(E_{i}\). Show that at least one of the sets \(E_{i}\) has measure

Show that there does not exist a Lebesgue measurable subset \(A \subset \mathbb {R}\) such that for every interval \((a, b) \subset \mathbb {R}\) we have \[ m(A \cap (a, b)) = \frac {b-a}{2} \] where \(m\) denotes Lebesgue measure.

Let \(f : [0, 1] \to \mathbb {R}\) be a continuous function and define a signed measure \(\mu \) on the Lebesgue measurable subsets of \([0, 1]\) by

where \(m\) is Lebesgue measure. Suppose that for all \(n = 0, 1, 2, 3, \ldots \)

Prove that \(\mu = 0\).

Let \(\mu \) be the outer measure on \(\mathbb {R}\) defined by \[ \mu (A) = \inf \left \{ \sum _{i=1}^{\infty }(b_{i} - a_{i}) \mid A \subseteq \bigcup _{i=1}^{\infty }(a_{i}, b_{i}],\ a_{i} \le b_{i} \right \} \]

for every \(A \subset \mathbb {R}\). Prove that \(\mu ((a, b]) = b - a\). You may use that the length of a segment covered by finitely many segments is less than or equal to the sum of the lengths of segments in the covering.

Let a measure space \((X, \mathcal {M}, \mu )\) be given and for a fixed measurable function \(f : X \to [0, \infty )\) define a new measure \(\lambda \) on \(\mathcal {M}\) by \[ \lambda (E) = \int _{E} f\,d\mu . \]

You don’t have to prove that \(\lambda \) is a measure. Prove that for any measurable \(g : X \to [0, \infty )\), \[ \int g\,d\lambda = \int gf\,d\mu . \]

Let \((X,\mathcal M)\) be a measurable space and \(\mu ,\nu \) two measures on it. Suppose that for every \(E\in \mathcal M\), \[ \mu (E)=0\quad \Rightarrow \quad \nu (E)=0. \] (a) Assuming \(\nu (X)<\infty \), prove that for every \(\epsilon >0\) there is \(\delta >0\) such that \[ \mu (E)<\delta \quad \Rightarrow \quad \nu (E)<\epsilon . \] You are not allowed to use the Radon-Nikodym theorem for this problem.

(b) Find an example where the above statement is false when \(\nu (X)=\infty \).

Let \((X,\mathcal M,\mu )\) be a measure space and let \(f:X\to [0,\infty )\) be a measurable function. Define a new measure \(\nu \) by \[ \nu (E)=\int _E f\,d\mu \] for \(E\in \mathcal M\) (you don’t have to prove that \(\nu \) is a measure). Prove that for every positive measurable function \(g\), \[ \int _X g\,d\nu =\int _X f g\,d\mu . \]

Let \((X,\mathcal M,\mu )\) be a measure space with \(\mu (X)<\infty \). Denote by \(S\) the set of equivalence classes of measurable functions \(X\to \mathbb R\) where \(f\sim g\) if and only if \(f=g\) a.e. For \(f,g\in S\) define \[ d(f,g)=\int _X\frac {|f(x)-g(x)|}{1+|f(x)-g(x)|}\,d\mu . \] Show that \((S,d)\) is a metric space and \(f_k\to f\) in \((S,d)\) if and only if \(f_k\to f\) in measure in \(X\).

Let \(\mathcal M\) be a \(\sigma \)-algebra of subsets of \(X\) and \(\mu \) a measure on \(\mathcal M\). Let \(A_1\subseteq A_2\subseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcup _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\).

Now assume that \(\mu (X)<\infty \), and let \(A_1\supseteq A_2\supseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcap _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\). Give an example where this fails if \(\mu (X)=\infty \).

Let \(f\ge 0\) be an integrable function on a measure space \(X\). Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f>\left (\int _X f\right )-\epsilon . \]

Let \(X=[0,1]\) with the usual measure. For \(n=1,2,\ldots \), let \(f_n=n^{1/p}\cdot \chi _{[0,1/n]}\). Prove that the sequence \(f_n\) does not converge weakly to \(0\) in \(L^1([0,1])\).

Let \(\mu \) be the outer measure on \(\mathbb R\) defined by \[ \mu (A)=\inf \left \{ \sum _{i=1}^{\infty }(b_i-a_i)\,\middle |\, A\subseteq \bigcup _{i=1}^{\infty }(a_i,b_i],\ a_i\le b_i \right \} \] for every \(A\subset \mathbb R\). Prove that \(\mu ((a,b])=b-a\). You may use that the length of a segment covered by finitely many segments is less than the sum of lengths of segments in the covering.

Let \((X,\mathcal M,\mu )\) be a measure space. Assume there exists \(\delta >0\) such that for every \(E\in \mathcal M\), either \(\mu (E)=0\) or \(\mu (E)>\delta \). Let \(f\in L^1(X)\). Prove that \(f\) is essentially bounded.

Let \(\mu \) be the outer measure on \(\mathbb R\) defined by \[ \mu (A)=\inf \left \{ \sum _{i=1}^{\infty }(b_i-a_i)\,\middle |\, A\subseteq \bigcup _{i=1}^{\infty }(a_i,b_i],\ a_i\le b_i \right \} \] for every \(A\subset \mathbb R\). Prove that \(\mu ((a,b])=b-a\). You may use that the length of a segment covered by finitely many segments is less than the sum of lengths of segments in the covering.

Let \((X,\mu )\) be a (not necessarily finite) measure space and \(f\ge 0\) be a \(\mu \) integrable function. Prove that, for every \(\epsilon >0\), there exists a set \(E\subset X\) of finite measure such that \[ \int _E f\,d\mu >\int _X f\,d\mu -\epsilon . \]

Let \(H=L^2(\mathbb R/\mathbb Z)\), with the usual Lebesgue measure on \(\mathbb R/\mathbb Z\) of total measure \(1\). It is well known that this space has a Hilbert basis \(e_n=e^{2\pi i n x}\), \(n\in \mathbb Z\). Let \(f\in H\) be given by \(f(x)=x\) on \([0,1)\). Compute \(\lVert f\rVert _2^2\) directly and then using Parseval’s equality.

Let \(X=[0,1]\) with the usual measure. Let \(p\ge 1\). For \(n=1,2,\ldots \), let \[ f_n=n^{1/p}\cdot \chi _{[0,1/n]}. \] Prove that the sequence \(f_n\) converges weakly to \(0\) in \(L^p([0,1])\) if and only if \(p>1\). Use that \[ L^p([0,1])^*\cong L^q([0,1]). \]

Let \(\mu \) be the outer measure on \(\mathbb R\) defined by \[ \mu (A)=\inf \left \{ \sum _{i=1}^{\infty }(b_i-a_i)\ \middle |\ A\subset \bigcup _{i=1}^{\infty }(a_i,b_i],\ a_i\le b_i \right \} \] for every \(A\subset \mathbb R\). Prove that \(\mu ((a,b])=b-a\). You may use that the length of a segment covered by finitely many segments is less than the sum of lengths of segments in the covering.

Prove or disprove: If \((X,\mathcal M,\mu )\) is a measure space, \(A_1,\ldots \) are measurable, and \[ \sum _{i=1}^{\infty }\mu (A_i)^2<\infty , \] then \[ \mu \left (\bigcap _{m=1}^{\infty }\bigcup _{i=m}^{\infty } A_i\right )=0. \]

This question considers different notions of convergence for \(\ell ^1(\mathbb N)\):

(a) Does pointwise convergence imply \(\lVert \cdot \rVert _1\) convergence? Vice versa?

(b) Does pointwise convergence imply convergence in measure? Vice versa?

Let \(\mu \) be a non-atomic Borel probability measure on \(\mathbb R\). Prove that there exists \(A\subset \mathbb R\) with \(\mu (A)=\frac 12\).

Let \((X,\mathcal M,\mu )\) be a measure space with \(\mu (X)<\infty \), and let \(1\le s<t\le \infty \). Show that \(L^t(X)\subset L^s(X)\).

Consider the interval \([-1,1]\) with a finite Borel measure \(\mu \) (not assumed to be the Lebesgue measure). Let \(C([-1,1])\) be the space of continuous functions endowed with the \(L^2\) norm, and let \(T:C([-1,1])\to \mathbb R\) be the operator \(T(f)=f(0)\). Show that \(T\) is a bounded operator (with respect to the \(L^2\) norm) if and only if \(\mu (\{0\})>0\).

Let \((X,\mathcal M,\mu )\) be a measure space, and suppose that \(A_n\in \mathcal M\), \(n=1,\ldots \), with \[ \sum _{n=1}^{\infty }\mu (A_n)<\infty . \] Show that the set of \(x\) belonging to infinitely many \(A_n\) has measure zero.

For \(f:\mathbb R\to \mathbb R\) and \(y\in \mathbb R\), define \(\tau _y f(x)=f(x-y)\) for \(x\in \mathbb R\). Assume that \(\mathbb R\) is equipped with the Lebesgue measure. For \(f\in L^1(\mathbb R)\), show that the map \(\mathbb R\to L^1(\mathbb R)\) defined by \(y\mapsto \tau _y f\) is uniformly continuous.

Prove that if \(\mu \) is a non-atomic Borel probability measure on \(\mathbb R\), then the function \(f:\mathbb R\to \mathbb R\) defined by \[ f(t)=\mu ((-\infty ,t)) \] is continuous.

For any subset \(A\) of \(\mathbb R\) we define its Lebesgue outer measure \[ \lambda ^*(A)=\inf \left \{ \sum _{i=0}^{\infty }|b_i-a_i| \;\middle |\; A\subset \bigcup _{i=0}^{\infty }(a_i,b_i) \right \}. \] Recall that given an outer measure \(\phi ^*:\mathcal P(X)\to [0,\infty ]\), we say a set \(A\) is \(\phi ^*\)-measurable if for all \(Y\in \mathcal P(X)\) we have \[ \phi ^*(Y)=\phi ^*(Y\cap A)+\phi ^*(Y\setminus A). \] Show that if \(\lambda ^*(\partial E)=0\), then \(E\) is \(\lambda ^*\)-measurable.

Let \((X,\mathcal B,\mu )\) be a measure space. \(U\in \mathcal B\) is called an atom if \(\mu (U)>0\) and \(\mu (V)=0\) for all \(V\), a proper subset of \(U\). If \((X,\mathcal B,\mu )\) is \(\sigma \)-finite measure space and \(f:X\to Y\) then \(f_*(\mu )\) has at most countably many atoms.

Let \((X,M,\mu )\) be a measure space, and suppose that \(1\le p<q<r\). Consider the vector space \(L^p\cap L^r\), with norm \(\lVert f\rVert =\lVert f\rVert _p+\lVert f\rVert _r\). Show that \(L^q\subset L^p\cap L^r\), and that the inclusion map \(i:L^q\to L^p\cap L^r\) is continuous.

Let \((X,M,\mu )\) be a measure space, and suppose that \(f:X\times [0,1]\to \mathbb R\) such that \(f(\cdot ,t):X\to \mathbb R\) is integrable for all \(t\). Define \[ F(t)=\int _X f(x,t)\,d\mu (x) \] for \(t\in [0,1]\). Suppose that \(\frac {\partial f}{\partial t}(x,t)\) exists for all \(x\) and \(t\), and that for some integrable function \(g\), \(\left |\frac {\partial f}{\partial t}(x,t)\right |\le g(x)\) for all \(x\) and \(t\). Show that \(F\) is differentiable and that \[ F'(t)=\int _X \frac {\partial f}{\partial t}(x,t)\,d\mu (x) \]

Let \((X,M,\mu )\) be a \(\sigma \)-finite measure space, and consider \(X\times X\) with the product measure. Suppose that \(K\in L^2(X\times X)\). Show that if \(f\in L^2(X)\), then \(Tf\in L^2(X)\) and \(\lVert Tf\rVert _2\le \lVert K\rVert _2\lVert f\rVert _2\) where \[ Tf(x)=\int _X K(x,y)f(y)\,d\mu (y). \]

Let \((X,\mathcal M,\mu )\) be a measure space, and let \(f\ge 0\) be an integrable function on \(X\). Suppose that \(f_n\ge 0\) is a measurable function, \(n=1,2,\ldots \), and that \[ f_1\le f_2\le \cdots \le f. \] If \(\int f_n\,d\mu \to \int f\,d\mu \), prove that \(f_n\to f\) a.e.

Let \((X,\mathcal M,\mu )\) be a measure space, \(1<p<\infty \), and \(q=p/(p-1)\). Suppose that \(f_n\in L^p(X)\), \(n=1,2,\ldots \), and that for each \(g\in L^q(X)\), \[ \sup _n \left |\int f_ng\,d\mu \right |<\infty . \] Show that \(\sup _n\lVert f_n\rVert _p<\infty \).

Recall that a topological space is said to be separable if it contains a countable dense set. Also recall that \(\ell ^p(\mathbb Z)\) is the \(L^p\)-space for counting measure on the integers. Prove or disprove the following statements:

(a) \(\ell ^1(\mathbb Z)\) is separable.

(b) \(\ell ^\infty (\mathbb Z)\) is separable.

Let \(f\) be a real-valued function on a measure space \(X\). Prove or disprove the following statements:

(a) If \(f\) is measurable, then so is \(|f|\).

(b) If \(|f|\) is measurable, then so is \(f\).

Let \(\mathbb R\) be equipped with Lebesgue measure. Suppose \(f\in L^1(\mathbb R)\) is uniformly continuous. Prove that \(\lim _{|x|\to \infty }f(x)=0\).

Let \((X,\mu )\) be a measure space, let \(\{f_n\}\) be a sequence of nonnegative integrable functions, and assume there exists an integrable function \(f\) such that \(f(x)=\lim _{n\to \infty } f_n(x)\) almost everywhere. Further assume \[ \tag {1} \lim _{n\to \infty }\int _X f_n\,d\mu =\int _X f\,d\mu . \] Prove that \[ \lim _{n\to \infty }\int _X |f_n-f|\,d\mu =0. \] Show by example that this need not hold if the assumption (1) is omitted.

Suppose \((X,\mathcal M,\mu )\) is a measure space and \(f:X\to \mathbb R\) is a real-valued function on \(X\). Suppose further that \(E_r:=\{x\mid f(x)>r\}\) is measureable for each rational number \(r\). Either prove the following assertion or find a counter-example: \(f\) is measureable.

Suppose \((X,\mathcal M,\mu )\) is a measure space and fix \(p\) and \(q\) finite such that \[ \frac {1}{p}+\frac {1}{q}=1. \] Let \(f_1,f_2,\ldots \) be a sequence of functions in \(L^p(X)\) converging (in \(L^p\)) to \(f\), and let \(g_1,g_2,\ldots \) be a sequence of functions in \(L^q\) converging (in \(L^q\)) to \(g\). Prove that the sequence \(f_1g_1,f_2g_2,\ldots \) converges to \(fg\) in \(L^1\). Does the same conclusion hold if \(p=1\) and \(q=\infty \)?

Suppose \((X,\mathcal M,\mu )\) is a measure space and that \(L^p(X)\subset L^q(X)\) for some \(1\le p<q<\infty \).

(a) Show that the inclusion \(L^p(X)\to L^q(X)\) is bounded.

(b) Let \(\mathcal M'\) denote the subsets in \(\mathcal M\) of nonzero measure. Use (a) to prove that \[ \inf _{E'\in \mathcal M'}\mu (E')>0. \]

(a) Let \(\ell ^1\) and \(\ell ^\infty \) be the Banach spaces of sequences of complex numbers \(\xi =(\xi _1,\xi _2,\ldots )\) such that \[ \xi \in \ell ^1 \quad \text {if and only if}\quad \lVert \xi \rVert _1:=\sum _{i=1}^{\infty }|\xi _i|<\infty \] and \[ \xi \in \ell ^\infty \quad \text {if and only if}\quad \lVert \xi \rVert _\infty :=\sup _i|\xi _i|<\infty . \] Let \((\ell ^\infty )^*\) denote the Banach space of continuous linear functionals, equipped with the operator norm, on \(\ell ^\infty \). Show that the map \(\Lambda :\ell ^1\to (\ell ^\infty )^*\) defined by \[ \Lambda (\xi )(\eta _1,\eta _2,\ldots )=\sum _i \xi _i\eta _i \] is a well-defined norm-preserving injection of \(\ell ^1\) into \((\ell ^\infty )^*\) which is not onto.

Hint: Show that \((\ell ^\infty )^*\) contains nontrivial functionals which vanish on the subspace consisting of those \(\xi \) such that \(\xi _i\to 0\) as \(i\to \infty \).

(b) Let \((X,\mu )\) be a measure space. State precise conditions on \(\mu \), \(p\), and \(q\) which guarantee a norm-preserving isomorphism of \(L^p(X)\) onto \((L^q(X))^*\).

Let \((X,\mathcal M,\mu )\) be a measure space such that \(\mu (X)<\infty \). Let \(f:X\to X\) be measure preserving in the sense that \(\mu (E)=\mu (f(E))\) for all \(E\in \mathcal M\). Prove that, for any \(E\in \mathcal M\), the set of points \(x\) of \(E\) such that the \(n\)th iterate \(f^n(x)\) is not in \(E\) for all \(n>0\) has zero measure.

Hint: Suppose not and obtain a contradiction with the fact that \(\mu (X)<\infty \).

Let \(\{f_n\}\), \(f\) be measurable functions on a measure space \((X,\mu )\) such that \[ f_1\ge f_2\ge \cdots \ge f\ge 0 \] Suppose that \(\int f_n\,d\mu \to \int f\,d\mu \) and that \(f_1\) is integrable. Prove that \(f_n\to f\) almost everywhere. Show by example, that this conclusion may be false if \(f_1\) is not integrable.

Let \((X,\mu )\) be a measure space with \(\mu (X)=1\), and let \(f\in L^\infty (X)\). Prove that \(\lVert f\rVert _p\to \lVert f\rVert _\infty \) where \(\lVert f\rVert _p\) denotes the \(L^p\) norm of \(f\).

Let \((\mathcal M,X,\mu )\) be a finite positive measure space.

(a) Show that \(L^2(X)\subset L^1(X)\), and that the inclusion \(i:L^2(X)\subset L^1(X)\) has norm \(\lVert i\rVert =\sqrt {\mu (X)}\).

(b) Let \(\mathcal M'\subset \mathcal M\) be the subset of measurable sets \(E\) with \(\mu (E)>0\). Show that if \[ L^1(X)=L^2(X), \] then \[ \inf _{E\in \mathcal M'}\mu (E)>0. \]

Let \((X,\mu )\) be a measure space and \(f\ge 0\) an integrable function on \(X\). Prove that for \(\epsilon >0\), there exists \(\delta >0\) such that whenever \(\mu (A)<\delta \), then \(\int _A f<\epsilon \).

Let \(f\) be integrable on \(\mathbb R\) with respect to Lebesgue measure, and define \[ F(x)=\int _{-\infty }^{x} f(t)\,dt \] Prove that \(F\) is continuous.

We say that \(f_n\to f\) in measure if for all \(\epsilon >0\) there exists an \(N>0\) such that if \(n\ge N\) then \[ \mu \bigl (\{x\mid |f_n(x)-f(x)|>\epsilon \}\bigr )<\epsilon . \] Show that if \(f_n\to f\) pointwise and \(\mu (X)<\infty \) then \(f_n\to f\) in measure.

Suppose \(X\) consists of two points \(a\) and \(b\). Define \(\mu (\{a\})=1\), \(\mu (\{b\})=\mu (X)=\infty \), and \(\mu (\emptyset )=0\). For this measure \(\mu \), prove or disprove: \(L^\infty (\mu )\) is the dual space of \(L^1(\mu )\).

Let \((\mathcal M,\mu ,X)\) be a positive measure space. Let \(\mathcal M'\) denote the collection of subsets (in \(\mathcal M\)) whose measures are finite. Prove that \[ L^2(X)\subset L^1(X) \] if and only if \[ \sup _{E\in \mathcal M'}\mu (E)<\infty . \]

Suppose \(f\) and \(g\) are two complex measurable functions on a measure space \(X\). Let \[ S=\{x\mid f(x)\ne g(x)\}. \] Prove or disprove: \(S\) is a measurable subset of \(X\).

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\in L^1(X)\). Let \(X_n=\{x\in X: |f(x)|>n\}\). Show that \[ \lim _{n\to \infty } n\mu (X_n)=0. \]

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\ge 0\) be a measurable function on \(X\). Define \(g(t)=\mu (\{x:f(x)\ge t\})\). Show that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty g(t)\,dt. \]

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) such that \(\mu (X)=1\). Let \(f\) be a bounded measurable function on \(X\). Show that \(f\in L^p(X)\) for \(1\le p<\infty \) and \(\|f\|_p\le \|f\|_q\) for \(1<p<q<\infty \).

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) and let \(f\) be a complex valued function on \(X\) which is integrable with respect to \(\mu \). For each \(n\), set \[ E_n=\{x\in X:1/n\le |f(x)|\le n\}. \] Then each \(E_n\) is a measurable set on which \(|f|\) is bounded. Prove that

(a) \(\mu (E_n)\) is finite for each \(n\); and

(b) \(\displaystyle \lim _{n\to \infty }\int _{E_n} f(x)\,d\mu (x) =\int _X f(x)\,d\mu (x)\).

Let \(f\) be holomorphic on the unit disc \(U\). Let \(h(z,w)\) be the function on \(U\times U\) defined by \[ h(z,w)= \begin {cases} \dfrac {f(z)-f(w)}{z-w} & \text {if } z\ne w,\\[0.6ex] f'(z) & \text {if } z=w \end {cases} \] Prove that \(f\) is continuous as a function of two variables on \(U\times U\). You need only check the continuity at points of the diagonal. At other points it is obvious.

Let \(X\) be a measure space with \(\mu (X)=1\). Let \(f\) be a measurable function in \(L^\infty (X)\). Note that \(\|f\|_p\le \|f\|_\infty \) for all \(p\ge 1\). In particular, \(f\) can be considered an element of \(L^p(X)\) for all \(p\ge 1\). Show that \[ \lim _{p\to \infty }\|f\|_p=\|f\|_\infty . \]

Let \(X\) be a compact Hausdorff space and denote by \(C(X)\) the Banach space of continuous complex valued functions on \(X\) and by \(P(X)\) the cone consisting of all \(f\in C(X)\) such that \(\operatorname {Re}(f)\ge 0\). If \(H\) is a linear subspace of \(C(X)\) such that \(H\cap P(X)=(0)\), prove that

(a) \(\|\lambda +h\|_\infty \ge |\lambda |\) for all \(\lambda \in \mathbb C\), \(h\in H\) (hint: show this is equivalent to \(\|1-h\|_\infty \ge 1\) for all \(h\in H\));

(b) there exists a positive Borel measure \(\mu \) on \(X\) such that \(\mu (X)=1\) and \[ \int h\,d\mu =0 \quad \text {for all } h\in H \] (recall that a measure \(\mu \) on \(X\) is positive if \(|\mu |=\mu (X)\)).

Let \(f\) be a non-negative measurable function and \(\mu \) a finite positive measure on a measure space \((X,\Sigma )\). Prove that \[ \lim _{n\to \infty }\int \frac {f^n(x)}{1+f^n(x)}\,d\mu (x) =\mu (E)+\frac {1}{2}\mu (F), \] where \(E=\{x\in X; f(x)>1\}\) and \(F=\{x\in X; f(x)=1\}\).

Let \(\mu \) be a measure on a measure space \(X\) and let \(f\ge 0\) be a measurable function on \(X\). Prove that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty \phi (y)\,dy \] where \(\phi (y)=\mu (\{x:f(x)\ge y\})\).

Let \(D\) be the closed unit disc in \(\mathbb C\) and \(T\) the unit circle. Prove that for every \(\mu \) positive measure of total mass one on \(D\) there is a positive measure \(\nu \) of total mass one on \(T\) such that \[ \int f\,d\mu =\int f\,d\nu \] for every \(f\) which is continuous on \(D\) and holomorphic on the interior of \(D\).

Let \(\mu \) be a measure on \(X\). Let \(f_1\ge f_2\ge \cdots f_n\ge \cdots \ge 0\) be a sequence of measurable functions. Assume \(\lim _{n\to \infty }\int _X f_n\,d\mu =0\). Show that \(f_n(x)\to 0\) a.e.

Let \(0<p<r<q\). Show that if \((X,\mathfrak M,\mu )\) is a (positive) measure space, then \[ L^r(\mu )\subset L^p(\mu )+L^q(\mu ); \] more precisely, for every \(f\in L^r(\mu )\) there exists \(g\in L^p(\mu )\) and \(h\in L^q(\mu )\) such that \(f=g+h\).

Let \((X,\mathfrak M,\mu )\) be a measure space, and let \(A_1\supset A_2\supset \cdots \supset A_n\supset \cdots \) be a sequence of sets in \(\mathfrak M\).

Is it always true that \(\mu (\bigcap _{n=1}^{\infty }A_n)=\lim _{n\to \infty }\mu (A_n)\)? If so, give a proof. Otherwise give a counterexample.

Let \(\mu \) be a positive finite measure on \((X,\mathfrak M)\) and \(g\in L^1(\mu )\). Suppose there exists \(0<M<\infty \) such that \[ \left |\int gs\,d\mu \right |\le M\lVert s\rVert _p\qquad (1\le p<\infty ) \] for all simple measurable functions on \(X\). Then show that \(g\in L^q(\mu )\) where \(\frac {1}{p}+\frac {1}{q}=1\).

Let \(\mathcal F=\{f_\alpha \}_{\alpha \in A}\) be a family of continuous complex valued functions on a compact Hausdorff space. Let \(\{c_\alpha \}_{\alpha \in A}\) be a family of complex numbers. Suppose that for each \(F\subset A\) with \(F\) finite, there exists a complex measure \(\mu _F\) of total variation one on \(X\) such that \[ \int _X f_\alpha \,d\mu _F=c_\alpha \qquad (\forall \alpha \in F). \] Show that there exists a complex measure \(\mu \) on \(X\) of total variation at most one such that \[ \int _X f_\alpha \,d\mu =c_\alpha \qquad (\forall \alpha \in A). \]

Let \((X,\mathfrak M,\mu )\) be a positive measure space. Let \(\psi \in L^2(\mu )\cap L^\infty (\mu )\). Show that \(\psi \in L^p(\mu )\) for all \(p\ge 2\).

Suppose that \((X,\mathfrak M,\mu )\) is a measure space with \(\mu \) a finite positive measure. Let \(I\) be an open interval in \(\mathbb R\) and \(f:X\times I\to \mathbb C\) a bounded function such that \(x\to f(x,t)\) is \(\mathfrak M\)-measurable for each \(t\in I\) and \(t\to f(x,t)\) is differentiable for each \(x\in X\). If \(\frac {d}{dt}f(x,t)\) is also bounded on \(X\times I\) prove that \(h(t)=\int _X f(x,t)\,d\mu (x)\) is differentiable on \(I\) and in fact \[ h'(t)=\int _X \frac {d}{dt}f(x,t)\,d\mu (x). \]

Let \((X,\mathcal M,\mu )\) be a measure space. Suppose \(\mu \) is finite and positive, and it satisfies the following property:

(*) for each \(A\in \mathcal M\), such that \(\mu (A)>0\), there exists \(B\in \mathcal M\) such that \(B\subset A\) and \(0<\mu (B)<\mu (A)\).

Prove that for each \(r>0\) there exists \(C\in \mathcal M\) such that \[ 0<\mu (C)<r. \]

Let \((X,\mathcal M,\mu )\) be a measure space, where the measure \(\mu \) is positive. Suppose \(f,f_n\in L^1(X,\mu )\), \(f_n\ge 0\) for each \(n\), \(f_n\to f\) a.e. and \[ \int _X f_n\,d\mu \to \int _X f\,d\mu \] Prove that \(f_n\to f\) in \(L^1(X,\mu )\) norm.

Hint: Show first that \((f-f_n)^+\le f\) for each \(n\) (recall that for real valued function \(g\), \(g^+=\max (g,0)\)).

Let \(X\) be a compact Hausdorff space.

(a) Prove that a complex measure \(\mu \) on \(X\) is positive if and only if \(\mu (X)=|\mu |(X)\).

(b) Let \(A\subset C(X)\) be a linear subspace containing the function \(1\). Let \(S\subset X\) be a compact subset with the property that for each \(f\in A\), \[ \sup \{|f(x)|:x\in X\}=\sup \{|f(x)|:x\in S\} \] Prove that for each \(x\in X\) there exists a positive measure \(\mu _x\) with support contained in \(S\) such that \[ f(x)=\int _S f\,d\mu _x\qquad \text {for }f\in A. \]