Measurable Functions

Let \(\mathcal {B}\) be the Borel \(\sigma \)-algebra on \(\mathbb {R}\) and \(\mathcal {M}\) be a \(\sigma \)-algebra on \(\mathbb {R}\). Prove that \(f:(\mathbb {R},\mathcal {M})\to (\mathbb {R},\mathcal {B})\) is measurable for all monotone functions \(f\) iff \(\mathcal {M}\) contains the Borel \(\sigma \)-algebra.

Suppose \(f_i:\mathbb {R}\to \mathbb {R}\) is measurable for \(i=1,2,\cdots \). Is the set \[ \{x\in \mathbb {R}\mid 2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \]

measurable? A number \(t\) is a limit point of a sequence if some subsequence converges to it.

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 \(X\) 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.

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 \(f:\mathbb {R}\to \mathbb {C}\) be Lebesgue measurable. Assume that \(f\) is invariant under translations by rational numbers. Show that \(f\) is constant Lebesgue almost everywhere.

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 \((X,\mathcal {M},\mu )\) 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 \(f : [0, 1] \to [0, \infty ]\) be measurable. Show that \(U(f) = \{(x, y) : 0 \le y \le f(x)\}\) is measurable and \[ \int f\,d\lambda = \lambda ^{2}(U(f)). \]

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 \((X, \mathcal {M})\) be a measurable space, so that \(\mathcal {M}\) is a \(\sigma \)-algebra of subsets of \(X\). Prove that \(\mathcal {M}\) is either finite or uncountable.

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

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 \(X\).

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 \[ \mu \left (\bigcap _{m=1}^{\infty }\bigcup _{i=m}^{\infty } A_i\right )=0. \]

Recall that (a version of) Lusin’s theorem states that if \(f:[0,1]\to \mathbb C\) is measurable then for all \(\epsilon >0\) there exists \(g:[0,1]\to \mathbb C\) continuous so that \[ \lambda (\{x\in [0,1]: g(x)\ne f(x)\})<\epsilon . \] Prove that Lusin’s Theorem implies that if \(f:\mathbb R\to \mathbb C\) is measurable then there exists a continuous function \(g:\mathbb R\to \mathbb C\) so that \[ \lambda (\{x:f(x)\ne g(x)\})\le 1. \]

If \(f:\mathbb R\to \mathbb R\) is Lipschitz and \(A\) is \(\lambda \)-measurable, then \(f(A)\) is measurable. Recall that a function is Lipschitz if there exists \(C\) so that \[ d(f(x),f(y))\le C\,d(x,y) \] for all \(x,y\in \mathbb R\).

Let \(f:\mathbb R^2\to \mathbb C\) be a Borel measurable function. Let \(g:\mathbb R^2\to \mathbb R^2\) be the map given by \(g(x,y)=(2x+y^2,y)\) for \((x,y)\in \mathbb R^2\). If \(f\in L^1(\lambda ^2)\), what is \[ \frac {\int _{\mathbb R^2}(f\circ g)\,d\lambda ^2} {\int _{\mathbb R^2}f\,d\lambda ^2}? \]

If \(f_i:\mathbb R\to \mathbb R\) are measurable functions, then is \[ \{x:2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \] measurable?

Recall that \(p\) is a limit point of a sequence \(\{a_i\}_{i=1}^{\infty }\) if there exists an increasing sequence of natural numbers \(n_i\) so that \(p=\lim _{i\to \infty }a_{n_i}\).

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 \(f\) 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 \(\{f_n\}\), \(f\) 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.

Decide whether the following statement is true or false. If true, prove it, if false, provide a counterexample.

Let \(\{f_n\}\) be a sequence of measurable functions (with respect to Lebesgue measure) with \(0\le f_n\le 1\) such that \(\int _0^1 f_n(x)\,dx\to 0\). Then \(f_n\to 0\) almost everywhere.

Assume that \(f\) is non-negative and \[ \int _X f\,d\mu <\infty . \] Show that for every \(\epsilon >0\) there exists a \(\delta >0\) such that if \(A\) is a measurable subset of \(X\) such that \(\mu (A)\le \delta \) then \[ \int _A f\,d\mu \le \epsilon . \]

Let \(\alpha \) be an irrational number. Let \(f\) be a real valued measurable function on \(\mathbb R\) such that \(f(x)=f(x+1)=f(x+\alpha )\). Use Fourier series to show that \(f\) is constant almost everywhere.

Let \(E_1,\ldots ,E_n\) by a finite collection of disjoint measurable subsets of \(X\) and let \(M\) be a subspace of \(L^2(X)\) spanned by the functions \(\{u_1,\ldots ,u_n\}\) where each \(u_i\) is in \(L^2(X)\) and has support contained in \(E_i\). Let \(P:L^2(X)\to \mathbb R\) be the orthogonal projection. Find a function \(h:X\times X\to \mathbb R\) such that \[ Pf(x)=\int _X h(x,y)f(y)\,d\mu (y) \] for all \(f\in L^2(X)\).

Let \(f\) and \(g\) be two real measurable functions on a measurable space \(X\). Show that \[ \{x\in X\mid \sin (f(x))\ge \exp (g(x))\} \] is measurable.

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 \(\{f_n\}\) be a sequence of complex valued measurable functions defined on a finite interval \((a,b)\subset \mathbb R\) (”measurable” means with respect to the Lebesgue measure). Suppose that there exists a constant \(M<\infty \) such that \(\int _a^b |f_n(x)|^2\,dx<M\) for all \(n\), and \(f_n(x)\to 0\) a. e. Prove that \[ \lim _{n\to \infty }\int _a^b f_n(x)\,dx=0. \] Show also that this assertion fails in general if one replaces the interval \((a,b)\) by \(\mathbb R\) (give a counterexample).

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\), and let \(f\ge 0\) be a function which is integrable with respect to \(\mu \). Let \(f_1\le f_2\le \cdots \le f_n\le \cdots \) be a sequence of real valued, measurable functions on \(X\), such that \(f_n\le f\) for each \(n\) and \(\int _X f_n\,d\mu \to \int _X f\,d\mu \). Show that \(f_n\to f\) almost everywhere.

Prove that a non-decreasing function from \(\mathbb R\) to \(\mathbb R\) is measurable with respect to the Borel \(\sigma \)-algebra.

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 \(f\) and \(g\) be two real valued measurable functions on a measurable space \(X\). Show that \[ \{x\in X:\sin f(x)\ge \cos g(x)\} \] is a measurable subset of \(X\).

Let \(f\) and \(g\) be two complex measurable functions on a measurable space \(X\). Show that \[ \{x\in X:f(x)\ne g(x)\} \] is a measurable subset of \(X\).

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 \(\mu \) 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 \(\frac {1}{p}+\frac {1}{q}=1\).