# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

4k views

### If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$

Let $1\leq p < \infty$. Suppose that $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite), $f_k \to f$ almost everywhere, and $\|f_k\|_{L^p} \to \|f\|_{L^p}$. Why ...
16k views

4k views

### Set of continuity points of a real function

I have a question about subsets $$A \subseteq \mathbb R$$ for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
14k views

### Does convergence in $L^{p}$ implies convergence almost everywhere?

If I know $\|f_{n}(x) - f(x)\|_{L^{p}(\mathbb{R})} \rightarrow 0$ as $n \rightarrow \infty$, do I know $\lim_{n \rightarrow \infty}f_{n}(x) = f(x)$ for almost every $x$?
5k views

### The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$

I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. ...
2k views

3k views

### Lebesgue Measure of the Graph of a Function

Let $f:R^n \rightarrow R^m$ be any function. Will the graph of $f$ always have Lebesgue measure zero? $(1)$ I could prove that this is true if $f$ is continuous. $(2)$ I suspect it is true if $f$ is ...
3k views

### Steinhaus theorem (sums version)

This is a question from Stromberg related to Steinhaus' Theorem: If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval. I can't quite see how to modify the ...
6k views

### Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?

I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of $P$...
4k views

### On the equality case of the Hölder and Minkowski inequalites

I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is the problem 4 of chapter 8. Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
2k views

### Is there a function with infinite integral on every interval?

Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
2k views

### Vitali-type set with given outer measure

Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion? Edit: I forgot to ...
3k views

### If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?

Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ ...
9k views

### Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the ...
9k views

### If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course and there is an exercise that I don't understand a part of his solution. The exercise is: Let $S$ be an infinite $\sigma$ algebra on $X$ .Prove ...
7k views

### Lebesgue measurable but not Borel measurable

I'm trying to find a set which is Lebesgue measurable but not Borel measurable. So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
2k views

### Continuous functions are differentiable on a measurable set?

I came across the following challenging problem in my self-study: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Then the set of points where $f$ is differentiable is a measurable set. I ...
2k views

### Preimage of generated $\sigma$-algebra

For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$. Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
2k views

### Is the image of a null set under a differentiable map always null?

Let $n$ be a positive integer. $\;$ Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be everywhere Frechet differentiable. Let $S$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero. Does it follow ...
523 views

### Continuity of $L^1$ functions with respect to translation

Let $f\in L^1$, consider the map $t\mapsto f_t=f(x-t)$, then how can one show that $t\mapsto f_t$ is continuous? More explicitly one wants to show that $\lim_{h\to 0}|f_{t+h}-f_t|_{L^1}=0$. I tried to ...
2k views

### Infinite product of measurable spaces

Suppose there is a family (can be infinite) of measurable spaces. What are the usual ways to define a sigma algebra on their Cartesian product? There is one way in the context of defining product ...
9k views

3k views

### Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets?

Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets? It is easy to prove this statement if the domain is bounded. Is there any way to extend the ...
638 views

### Summing over General Functions of Primes and an Application to Prime $\zeta$ Function

Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following: $$\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}$$ and ...
2k views

8k views

### Under what condition we can interchange order of a limit and a summation?

Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? ...
1k views

2k views

### Does an absolutely integrable function tend to $0$ as its argument tends to infinity?

Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that $$\int_{0}^\infty|f(t)|dt<\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$ If so can you provide a proof, ...
191 views

### Show that $\lim _{r \to 0} \|T_rf−f\|_{L_p} =0.$

I am having a hard time with the following real analysis qual problem. Any help would be awesome. Thanks. Suppose that $f \in L^p(\mathbb{R}),1\leq p< + \infty.$ Let $T_r(f)(t)=f(t−r).$ Show ...
Let $\mathcal{L}$ denote the $\sigma$-algebra of Lebesgue measurable sets on $\mathbb{R}$. Then, if memory serves, there is an example (and of course, if there is one, there are many) of a continuous ...