Department of Mathematics University of Utah Real and Complex Analysis Preliminary Examination May 2020

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 \(C([0,1])\) be the Banach space of continuous functions on \([0,1]\) with the sup norm and view it as a natural subspace of \(L^\infty ([0,1])\), where \([0,1]\) is given Lebesgue measure, and the norm is essential supremum. Prove the existence of a bounded functional on \(L^\infty ([0,1])\) which is not identically \(0\) but vanishes on \(C([0,1])\).

Let \(f_n\in C([0,1])\) for \(n=1,2,\ldots \). Show that the following statements are equivalent:

(a) For every \(\lambda \in C([0,1])^*\) we have \(\lambda (f_n)\to 0\) as \(n\to \infty \).

(b) \(f_n(x)\to 0\) for every \(x\in [0,1]\) and \(\sup \lVert f_n\rVert _\infty <\infty \).

Here \(\lVert f\rVert _\infty =\sup _{x\in [0,1]}|f(x)|\).

Let \(\mathcal H\) be a Hilbert space and \(P:\mathcal H\to \mathcal H\) a self-adjoint operator, i.e. \(\langle Px,y\rangle =\langle x,Py\rangle \) for all \(x,y\in \mathcal H\). Also assume that \(P^2=P\). Prove that \(P\) is orthogonal projection to a closed subspace. Note that we are not assuming that \(P\) is bounded.

Let \(f\) be an analytic function. Show that the successive derivatives of \(f\) at a point can never satisfy \[ |f^{(n)}(z)|>n!n^n \] for all \(n\).

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x^{1/3}}{1+x^2}\,dx. \]

If \(f(z)\) is analytic in \(|z|\le 1\) and satisfies \(|f|=1\) on \(|z|=1\), show that \(f(z)\) is rational.

Show that the family of functions \(\{z^n\}_{n\in \mathbb Z_{\ge 0}}\) form a normal family in \(|z|<1\), also in \(|z|>1\), but not in any region that contains a point on the unit circle.

Let \(\wp (z)\) be the Weierstrass \(\wp \) function. Prove that there are constants \(g_2\) and \(g_3\) such that \[ \wp '(z)^2=4\wp (z)^3-g_2\wp (z)-g_3. \]