Greatest Hits

\(\star\): Show that for \(E\subseteq {\mathbf{R}}^n\), 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 \(E\subseteq {\mathbf{R}}^n\) 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 \({\mathbf{R}}^s\), then \(F(x, y) \coloneqq f(x)\) is measurable on \({\mathbf{R}}^s\times{\mathbf{R}}^t\) 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} \intf(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 \(X\subseteq {\mathbf{R}}\) 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 \(f\in C^1({\mathbf{R}})\) 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
 \(f_n: [a, b]\to {\mathbf{R}}\) 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 \({\mathbf{R}}\).
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 \(E\subseteq {\mathbf{R}}^n\) 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({\mathbf{Z}}) &\subset \ell^1({\mathbf{Z}}) \subset \ell^\infty({\mathbf{Z}}) .\end{align*}
Unsorted
If \(\left\{{R_j}\right\} \rightrightarrows R\) is a covering of \(R\) by rectangles, \begin{align*} R = \overset{\circ}{\coprod_{j}} R_j &\implies {\left\lvert {R} \right\rvert} = \sum {\left\lvert {R} \right\rvert}_j \\ R \subseteq \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 \(U \subseteq {\mathbf{R}}\) is a countable union of disjoint open intervals.
 Show that every open \(U \subseteq {\mathbf{R}}^n\) is a countable union of almost disjoint closed cubes.
 Show that that Cantor middlethirds set is compact, totally disconnected, and perfect, with outer measure zero.
 Prove the BorelCantelli lemma.