querent

Let $𝑈\subset {ℝ}^2$ be the open set

For which $𝑎>0$ does the integral

converge, where

Suppose $𝐸\subset ℝ$ is a (Lebesgue) measurable subset. For all $𝑥\in ℝ$, define

Prove that for a.e. $𝑥\in 𝐸$,

Find the flux of the vector field

in ${ℝ}^3$ through the surface of the unit sphere, oriented by the outward-pointing normal.

Show that the mapping

is locally invertible at every point $(𝑥,𝑦)\in {ℝ}^2$.

Show that the set $\mathbb{N}$ of natural numbers can be represented as a union of uncountably many distinct subsets

in such a way that for any $𝛼,𝛽\in 𝐴$, either ${𝑁}_{𝛼}={𝑁}_{𝛽}$, or the intersection ${𝑁}_{𝛼}\cap {𝑁}_{𝛽}$ is finite.

Let $𝑓,𝑔\in {𝐿}^2({ℝ}^{𝑛})$. Show that $𝑓\ast 𝑔\in {𝐶}^0({ℝ}^{𝑛})$ and ${lim}_{|𝑥|\to \infty }(𝑓\ast 𝑔)(𝑥)=0$.

Fix $1\le 𝑝<𝑞<\infty$.

(a) Suppose $𝑓\in {𝐿}^{𝑝(ℝ)}$ and ${\int }_{𝐴}{|𝑓|}^{𝑞}<\infty$ for any measurable $𝐴$ with $|𝐴|<\infty$. Show that $𝑓\in {𝐿}^{𝑞(ℝ)}$.

(b) Find a counterexample if we only assume ${\int }_{𝑎}^{𝑎+1}{|𝑓|}^{𝑞}<\infty$ for all $𝑎\in ℝ$.

Let $(𝑋,𝐴,𝜇)$ be a measure space and let $𝑓\in {𝐿}^1(𝑋,𝜇)$. Show that

Let $𝑋$ be a Banach space over $ℝ$.

(a) Let $𝑌\subset 𝑋$ be a subspace. Show that $𝑌$ is closed in the norm topology if and only if $𝑌$ is closed in the weak topology.

(b) Suppose now that $𝑌\subset 𝑋$ is a subset that is not necessarily a subspace. Is it still the case that $𝑌$ is closed in the norm topology if and only if $𝑌$ is closed in the weak topology? Prove it or give a counterexample.

Suppose that ${\{{𝑓}_{𝑛}\}}_{𝑛=1}^{\infty }$ is a sequence of measurable functions such that ${𝑓}_{𝑛}\to 0$ almost everywhere on $[0,1]$. Prove that there exists a sequence ${𝑐}_{𝑛}$ such that $|{𝑐}_{𝑛}|\to \infty$ as $𝑛\to \infty$ and ${𝑐}_{𝑛}{𝑓}_{𝑛}\to 0$ almost everywhere.