# 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.

• 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

If $$\left\{{R_j}\right\} \rightrightarrows R$$ is a covering of $$R$$ by rectangles, \begin{align*} R = \overset{\circ}{\displaystyle\coprod_{j}} R_j &\implies {\left\lvert {R} \right\rvert} = \sum {\left\lvert {R} \right\rvert}_j \\ R \subseteq \displaystyle\bigcup_j R_j &\implies {\left\lvert {R} \right\rvert} \leq \sum {\left\lvert {R} \right\rvert}_j .\end{align*}

• 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.