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4 votes
Accepted

Understanding the proof of $L^{\infty}$ is complete.

To answer this, the following statement is true: Let $(f_n), (g_n)$ be two sequences in $L^p$ such that $f_n = g_n$ a.e, $f_n \rightarrow f$ a.e, then $g_n \rightarrow f$ a.e Indeed, let $A_n = \{...
Thành Nguyễn's user avatar
3 votes

Understanding the proof of $L^p(X,\mathscr{A},\mu)$ is complete ($1\leq p<+\infty$)

I might look at your solutions to the other points tomorrow. Very briefly: You can notice $(4)$ and $(2)$ give $(5)$. $(1),(2),(3)$ all follow just from the explicit estimates you have in the post and ...
FShrike's user avatar
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2 votes

Understanding the proof of $L^{\infty}$ is complete.

I really appreciate @ThànhNguyễn's answer! This answer post is for my own record with extra details worked out. If you find any mistakes, please comment below! Thanks a lot! Let $\{f_k\}$ be a ...
Beerus's user avatar
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1 vote
Accepted

Integral function of bounded variation function derivative

Extend $f$ to all of $\mathbb{R}$ by setting $f(x) = f(a)$ for $x < a$ and $f(x) = f(b)$ for $x > b$. Then by the Fatou's lemma, \begin{align*} \int_{a}^{b} |f'(x)| \, \mathrm{d}x &=\int_{a}^...
Sangchul Lee's user avatar
1 vote
Accepted

Given $f \in L^p_{\text{loc}}(\Omega) \setminus L^\infty(\Omega)$, does it follow that $A \cap S(f,K) \neq \emptyset$ for all $K > 0$?

This answer just replicates my comment: For any two sets $A,B$ we can write $B$ as a disjoint union $$ B = (B\cap A) \cup (B \cap A^c)$$ If we know the sets $A,B$ are Lebesgue measurable we know $A^c$,...
Michael's user avatar
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