Extra Problems: Measure Theory

Greatest Hits

• \(\star\): Show that for \(E\subseteq {\mathbb{R}}^n\), TFAE:

  • \(E\) is measurable
  • \(E = H\cup Z\) here \(H\) is \(F_\sigma\) and \(Z\) is null
  • \(E = V\setminus Z'\) where \(V\in G_\delta\) and \(Z'\) is null.

• \(\star\): Show that if \(E\subseteq {\mathbb{R}}^n\) is measurable then \(m(E) = \sup \left\{{ m(K) {~\mathrel{\Big\vert}~}K\subset E\text{ compact}}\right\}\) iff for all \({\varepsilon}> 0\) there exists a compact \(K\subseteq E\) such that \(m(K) \geq m(E) - {\varepsilon}\).

• \(\star\): Show that cylinder functions are measurable, i.e. if \(f\) is measurable on \({\mathbb{R}}^s\), then \(F(x, y) \coloneqq f(x)\) is measurable on \({\mathbb{R}}^s\times{\mathbb{R}}^t\) for any \(t\).

• \(\star\): Prove that the Lebesgue integral is translation invariant, i.e. if \(\tau_h(x) = x+h\) then \(\int \tau_h f = \int f\).

• \(\star\): Prove that the Lebesgue integral is dilation invariant, i.e. if \(f_\delta(x) = {f({x\over \delta}) \over \delta^n}\) then \(\int f_\delta = \int f\).

• \(\star\): Prove continuity in \(L^1\), i.e. \begin{align*} f \in L^{1} \Longrightarrow \lim _{h \rightarrow 0} \int|f(x+h)-f(x)|=0 .\end{align*}

• \(\star\): Show that \begin{align*}f,g \in L^1 \implies f\ast g \in L^1 {\quad \operatorname{and} \quad} {\left\lVert {f\ast g} \right\rVert}_1 \leq {\left\lVert {f} \right\rVert}_1 {\left\lVert {g} \right\rVert}_1.\end{align*}

• \(\star\): Show that if \(X\subseteq {\mathbb{R}}\) with \(\mu(X) < \infty\) then \begin{align*} {\left\lVert {f} \right\rVert}_p \overset{p\to\infty}\to {\left\lVert {f} \right\rVert}_\infty .\end{align*}

Topology

• Show that every compact set is closed and bounded.
• Show that if a subset of a metric space is complete and totally bounded, then it is compact.
• Show that if \(K\) is compact and \(F\) is closed with \(K, F\) disjoint then \(\operatorname{dist}(K, F) > 0\).

Continuity

• Show that a continuous function on a compact set is uniformly continuous.

Differentiation

• Show that if \(f\in C^1({\mathbb{R}})\) and both \(\lim_{x\to \infty} f(x)\) and \(\lim_{x\to \infty} f'(x)\) exist, then \(\lim_{x\to\infty} f'(x)\) must be zero.

Advanced Limitology

• If \(f\) is continuous, is it necessarily the case that \(f'\) is continuous?
• If \(f_n \to f\), is it necessarily the case that \(f_n'\) converges to \(f'\) (or at all)?
• Is it true that the sum of differentiable functions is differentiable?
• Is it true that the limit of integrals equals the integral of the limit?
• Is it true that a limit of continuous functions is continuous?
• Show that a subset of a metric space is closed iff it is complete.
• Show that if \(m(E) < \infty\) and \(f_n\to f\) uniformly, then \(\lim \int_E f_n = \int_E f\).

Uniform Convergence

• Show that a uniform limit of bounded functions is bounded.
• Show that a uniform limit of continuous function is continuous.
  • I.e. if \(f_n\to f\) uniformly with each \(f_n\) continuous then \(f\) is continuous.
• Show that
  • \(f_n: [a, b]\to {\mathbb{R}}\) are continuously differentiable with derivatives \(f_n'\)
  • The sequence of derivatives \(f_n'\) converges uniformly to some function \(g\)
  • There exists at least one point \(x_0\) such that \(\lim_n f_n(x_0)\) exists,
  • Then \(f_n \to f\) uniformly to some differentiable \(f\), and \(f' = g\).
• Prove that uniform convergence implies pointwise convergence implies a.e. convergence, but none of the implications may be reversed.
• Show that \(\sum {x^n \over n!}\) converges uniformly on any compact subset of \({\mathbb{R}}\).

Measure Theory

• Show that continuity of measure from above/below holds for outer measures.

• Show that a countable union of null sets is null.

Measurability

• Show that \(f=0\) a.e. iff \(\int_E f = 0\) for every measurable set \(E\).

Integrability

• Show that if \(f\) is a measurable function, then \(f=0\) a.e. iff \(\int f = 0\).
• Show that a bounded function is Lebesgue integrable iff it is measurable.
• Show that simple functions are dense in \(L^1\).
• Show that step functions are dense in \(L^1\).
• Show that smooth compactly supported functions are dense in \(L^1\).

Convergence

• Prove Fatou’s lemma using the Monotone Convergence Theorem.
• Show that if \(\left\{{f_n}\right\}\) is in \(L^1\) and \(\sum \int {\left\lvert {f_n} \right\rvert} < \infty\) then \(\sum f_n\) converges to an \(L^1\) function and \begin{align*}\int \sum f_n = \sum \int f_n.\end{align*}

Convolution

• Show that if \(f, g\) are continuous and compactly supported, then so is \(f\ast g\).
• Show that if \(f\in L^1\) and \(g\) is bounded, then \(f\ast g\) is bounded and uniformly continuous.
• If \(f, g\) are compactly supported, is it necessarily the case that \(f\ast g\) is compactly supported?
• Show that under any of the following assumptions, \(f\ast g\) vanishes at infinity:
  • \(f, g\in L^1\) are both bounded.
  • \(f, g\in L^1\) with just \(g\) bounded.
  • \(f, g\) smooth and compactly supported (and in fact \(f\ast g\) is smooth)
  • \(f\in L^1\) and \(g\) smooth and compactly supported (and in fact \(f\ast g\) is smooth)
• Show that if \(f\in L^1\) and \(g'\) exists with \({\frac{\partial g}{\partial x_i}\,}\) all bounded, then \begin{align*}{\frac{\partial }{\partial x_i}\,}(f\ast g) = f \ast {\frac{\partial g}{\partial x_i}\,}\end{align*}

Fourier Analysis

• Show that if \(f\in L^1\) then \(\widehat{f}\) is bounded and uniformly continuous.
• Is it the case that \(f\in L^1\) implies \(\widehat{f}\in L^1\)?
• Show that if \(f, \widehat{f} \in L^1\) then \(f\) is bounded, uniformly continuous, and vanishes at infinity.
  • Show that this is not true for arbitrary \(L^1\) functions.
• Show that if \(f\in L^1\) and \(\widehat{f} = 0\) almost everywhere then \(f = 0\) almost everywhere.
  • Prove that \(\widehat{f} = \widehat{g}\) implies that \(f=g\) a.e.
• Show that if \(f, g \in L^1\) then \begin{align*}\int \widehat{f} g = \int f\widehat{g}.\end{align*}
  • Give an example showing that this fails if \(g\) is not bounded.
• Show that if \(f\in C^1\) then \(f\) is equal to its Fourier series.

Approximate Identities

• Show that if \(\phi\) is an approximate identity, then \begin{align*}{\left\lVert {f\ast \phi_t - f} \right\rVert}_1 \overset{t\to 0}\to 0.\end{align*}
  • Show that if additionally \({\left\lvert {\phi(x)} \right\rvert} \leq c(1 + {\left\lvert {x} \right\rvert})^{-n-{\varepsilon}}\) for some \(c,{\varepsilon}>0\), then this converges is almost everywhere.
• Show that is \(f\) is bounded and uniformly continuous and \(\phi_t\) is an approximation to the identity, then \(f\ast \phi_t\) uniformly converges to \(f\).

\(L^p\) Spaces

• Show that if \(E\subseteq {\mathbb{R}}^n\) is measurable with \(\mu(E) < \infty\) and \(f\in L^p(X)\) then \begin{align*}{\left\lVert {f} \right\rVert}_{L^p(X)} \overset{p\to\infty}\to {\left\lVert {f} \right\rVert}_\infty.\end{align*}
• Is it true that the converse to the DCT holds? I.e. if \(\int f_n \to \int f\), is there a \(g\in L^p\) such that \(f_n < g\) a.e. for every \(n\)?
• Prove continuity in \(L^p\): If \(f\) is uniformly continuous then for all \(p\), \begin{align*}{\left\lVert {\tau_h f - f} \right\rVert}_p \overset{h\to 0}\to 0.\end{align*}
• Prove the following inclusions of \(L^p\) spaces for \(m(X) < \infty\): \begin{align*} L^\infty(X) &\subset L^2(X) \subset L^1(X) \\ \ell^2({\mathbb{Z}}) &\subset \ell^1({\mathbb{Z}}) \subset \ell^\infty({\mathbb{Z}}) .\end{align*}

Unsorted

• Show that any disjoint intervals is countable.
• Show that every open \(U \subseteq {\mathbb{R}}\) is a countable union of disjoint open intervals.
• Show that every open \(U \subseteq {\mathbb{R}}^n\) is a countable union of almost disjoint closed cubes.
• Show that that Cantor middle-thirds set is compact, totally disconnected, and perfect, with outer measure zero.
• Prove the Borel-Cantelli lemma.