# Tagged Questions

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

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### 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 ...
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### 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 $x$?
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### 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 $\mathcal{A}$. ...
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### 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 ...
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### 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 ...
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### 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 $P$...
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### 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 ...
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### 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?
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### 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 ...
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### 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 $\|fg\|_1 \leq C\|g\|_q$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Y$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Lebesgue measurable set that is not a Borel measurable set

exact duplicate of Lebesgue measurable but not Borel measurable BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck... In short: Is there a Lebesgue ...
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### The construction of a Vitali set

Can anyone explain the concept of a Vitali set? I am not able to understand the construction of the set.
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### Measure of the Cantor set plus the Cantor set

The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets: If $C$ ...
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### Limit of measures is again a measure

Given a sequence $(\mu_n)_{n\in \mathbb N}$ of finite measures on the measurable space $(\Omega, \mathcal A)$ such that for every $A \in \mathcal A$ the limit $$\mu(A) = \lim_{n\to \infty} \mu_n(A)$$...
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### Positive outer measure set and nonmeasurable subset

I'm attending a course of Measure and Integration and have some homework to do. We don't have a specific book to follow, neither for exercise. I'm asked to proof that every set $A \in R$ with ...
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### Understanding proof of completeness of $L^{\infty}$ [closed]

I'm reading page number 4 here. In particular the section where it deals with the case $p=\infty$, that is , showing that $L^{\infty}$ is complete. http://www.core.org.cn/NR/rdonlyres/Mathematics/18-...
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### The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra

The union of a sequence of $\sigma$-algebras need not be a $\sigma$-algebra, but how do I prove the stronger statement below? Let $\mathcal{F}_n$ be a sequence of $\sigma$-algebras. If the ...
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### Formal definition of conditional probability

It would be extremely helpful if anyone gives me the formal definition of conditional probability and expectation in the following setting, given probability space $(\Omega, \mathscr{A}, \mu )$ ...
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I'm trying to prove the following theorem: Let $\{f_n\}\subset L^p(\Omega)$, $f_n \rightharpoonup f$ in $L^p(\Omega)$ ($\Omega\subset\mathbb{R}^n$ is open and bounded, $1\leq p \leq \infty$) and $f_n ... 1answer 1k 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, ... 3answers 188 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 ... 3answers 9k views ### Example where union of increasing sigma algebras is not a sigma algebra If$\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$are sigma algebras, what is wrong with claiming that$\cup_i\mathcal{F}_i$is a sigma algebra? It seems closed under complement since for all$...
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 ...