For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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1answer
103 views

Absolute value bound of Lebesgue integral

For the Riemann integral, we have the bound $$\left|\int_Af(x)dx\right|\leq\left(\sup_{x\in A}|f(x)|\right)\cdot\left|\int_Adx\right|$$ Do we have a similar bound for the Lebesgue integral, one like ...
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3answers
180 views

Why is expectation defined by $\int xf(x)dx$?

I recently found out that the expectation of a random variable $X$ in a probability space $(\Omega, \mathcal F, \mathbb P)$, $\mathbb E(X)$, is just the term used in probability theory for the ...
2
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2answers
118 views

Integral of Schwartz function over probability measure

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that ...
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1answer
39 views

Integral of exponent of random variable is continuous

Let $X$ be a set, $F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given a random variable $f:X\rightarrow\mathbb{R}$, define $$\chi_f(t)=\int_Xe^{itf}d\mu$$ Show ...
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2answers
44 views

Fix some $\delta\in \mathbb R$ and let $f:[0,\infty)\rightarrow \mathbb R$ be given by the equation

$$f(x)=\frac{\sin(x^2)}{x}+\frac{\delta x}{1+x}$$ Show that, $\lim_{n\rightarrow\infty}\int_{0}^{a}f(nx)\ dx=a\delta$ for each $\ a>0$. My attempt: $\lim_{n\rightarrow\infty}\ f(nx)=\delta$ ...
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1answer
58 views

Lebesgue integrability of $\ f$ and $\ f^{-1}$

Suppose $\ f:\ X\rightarrow(0,\infty)$ is a measurable function. If $$\int_{X} f\ d\mu<\infty\ $$ and $$\int_{X} \dfrac1f\ d\mu<\infty $$ Show that $\mu(X)<\infty$.
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1answer
112 views

Find a Borel subset satisfying the condition

Let $\alpha \in (0,1)$. Find a fixed Borel subset $E$ of $[-1,1]$ such that $$ \lim_{r \rightarrow 0^+} \frac{m(E \cap [-r,r])}{2r} = \alpha $$ I think it is the trickiest problem for studying ...
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1answer
55 views

How to apply Fubini here

Let $f$ and $g$ be integrable functions on the measure space $(X, \mathcal M, \mu)$ with the property that $$ \mu(\{f > t\} \triangle \{g > t\}) = 0 $$ for $\lambda$-a.e. $t \in \mathbb R$. ...
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0answers
120 views

If $\int(f_n) \rightarrow \int(f)$ then $\int(|f_n-f|) \rightarrow 0$ for $f_n \rightarrow f$ pointwise

I'd like to show that for an integrable sequence of functions $f_n:X \rightarrow [0, \infty)$ with $\sup_{n\geq 1} \int_{X} f_n d\mu < \infty, f_n \rightarrow f$ pointwise a.e. for some function ...
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1answer
137 views

A measurable function with $\int f^n$ bounded or converging as $n \to \infty$

(1) Show that, if $f^n$ is integrable for all integers $n\ge 1$ and $\limsup_{n\to \infty} \int f^n<\infty$ then $|f|\le1$ almost everywhere. (2) Show that, if $f^n$ is integrable for ...
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2answers
71 views

Understanding the last few lines in a proof by Royden in Real Analysis.

The book states: $\textbf{Proposition 2}$: Let $C$ be a countable subset of the open interval $(a, b)$. Then there is an increasing function on $(a, b)$ that is continuous only at points in $(a, b) ...
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0answers
23 views

norm of $x \in \mathbb R^d$ is in Sobolev space

For which values of $\alpha, k,p,d$ is $$ \|x\|^\alpha \in \textrm{W}^{k,p} (B(0,1)) \quad ? $$ where $\displaystyle{ \textrm{B}(0,1) = \{x \in \mathbb R^d : \|x\|<1\}}$ This is an ...
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2answers
288 views

Need help badly: n-dimensional Lebesgue measure of a hyperplane is zero.

Let $\alpha \in \mathbb{R}$, $a \neq 0$, and $\mu \in \mathbb{R}^{n}$. Let $H$ be the hyperplane in $R^{n}$ given by $h = \{ x \in \mathbb{R}^{n} : \langle x-\mu , a \rangle = 0 \}$. Show that ...
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2answers
215 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
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1answer
51 views

$L^2$ function on finite interval implies $L^1$?

Let $a,b\in\mathbb{R}$. Suppse $f:\mathbb{R}\rightarrow\mathbb{C}$ is an $L^2$ function on the finite interval $(a,b)$. That is, $$\int_{a}^b|f(x)|^2dx<\infty$$ Is it always true that $f$ is an ...
3
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1answer
61 views

convergence on $L^p$ space

Let $ \displaystyle{ f \in L^p (\mathbb R^n), 1\leq p <\infty }$ and let $ \upsilon \in \mathbb R^n$. For $h>0$ define $\displaystyle{ f_h(x) = \frac{1}{h} \int_0^h f(x+s \upsilon) ds }$. ...
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1answer
61 views

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$) Here's a theorem from my real analysis book: Assume $1\le p\le \infty$ and ...
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1answer
106 views

step function vs integrable function

The Problem: Let f be an integrable function on $\mathbb{R}$ with Lebesgue measure. Thern for any $\epsilon >0$ there is a finite collection of intervals $E_1, . . . , E_k$ (that is, $E_i = [a_i, ...
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1answer
165 views

On why the Vitali Covering Lemma does not apply when the covering collection contains degenerate closed intervals

I believe I have a fundamental misunderstanding of the concept of the Vitali Covering Lemma. Definition - A closed bounded interval $[c, d]$ is said to be nondegenerate provided $c < d$. ...
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69 views

Does showing a function is integrable suffice to show the function is measurable?

I am reviewing past homework exercises in preparation for a midterm exam. Fortunately, my professor provides solutions. However, I found one of his solutions contains an (seemingly) important ...
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1answer
108 views

Show a set is in Borel sigma algebra

First of all, sorry for asking again a question. For all functions, $f:[a,b] \to \mathbb{R}_{+} $ define $S(f)=\{(x,y)\in \mathbb{R}^{2}: 0\leq y \leq f(x)\}$. Show if $f$ is measurable, then ...
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1answer
210 views

Properties of $||f||_{\infty}$ - the infinity norm

Prove that $||f||_{\infty}$ is the smallest of all numbers of the form $\sup\{|g(x)|: x\in X\}$, where $f=g$ ($\mu$ almost everywhere). In addition, if $f$ is a continuous function on the measure ...
2
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1answer
108 views

Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$

Let $1<p_0<\infty$. Find an example of a sequence $\{f_k\}$ such that $f_k\in L^p$ for $1\le p <\infty$, $f_k\to0$ in $L^p$ for $1\le p < p_0$, but $f_k$ does not converge in $L^{p_0}$. ...
2
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3answers
77 views

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$.

Suppose $\{f_k\}$ is a sequence of $M$-measurable functions on $X$. Let $p_1$ and $p_2\in [1,\infty)$, and suppose $f_k\in L^{p_1}\cap L^{p_2}$. Also suppose there exist $g\in L^{p_1}$ and $h\in ...
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1answer
160 views

Proving a few things about $ L^{p} $-spaces

I am new to $ L^{p} $-spaces and am trying to prove a few things about them. Therefore, I would like to ask you whether I have gotten the following right. Prove that $ {L^{\infty}}(I) \subseteq ...
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2answers
124 views

Why is the undergraph definition of Lebesgue integral so rare?

So in Pugh's Real Mathematical Analysis, the initial definition of the Lebesgue integral is as the Lebesgue measure of the undergraph of the function (where the function is nonnegative, with the usual ...
2
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1answer
41 views

Prove under finite measure equivalent statements

Prove that, if $\mu(\Omega)<\infty$, $(f_n)\subset L^1(\Omega,\mu)$ and $\int_{\Omega}|f_n|d\mu\leq M<\infty$, then ...
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1answer
101 views

If a function is differentiable almost everywhere, can it be written as an integral?

Consider a function $f:\mathbb{R}^n \to \mathbb{R}$. If $f$ is differentiable with Lebesgue integrable derivative, we may write $$ f(x+y) - f(x) = \sum_{i=1}^p \int_0^1 y_i \nabla f_i(x+ty)dt $$ by ...
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1answer
117 views

Can I use this trick for my proof?

I am given the following problem, with a hint that states that I should use the General Lebesgue Dominated Convergence Theorem. Let $\{f_n\}$ be a sequence of integrable functions on $E$ for which ...
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1answer
172 views

total variation of continuous differentiable function

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function, differentiable on $(0,1)$ and such that $\,f'$ is continuous on $(0,1)$. Prove that $f$ is of bounded variation and $$TV(f,[0,1]) = ...
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3answers
166 views

Dominated convergence theorem arctan(nx)

Calculate $$\lim\limits_{n\to\infty}\int_a^{\infty}\frac{n}{1+n^2x^2}\,d\mu$$ where $\mu$ is the Lebesgue measure and $a\geq0$. First is easy to see that $\arctan(nx) ' = \frac{n}{1+n^2x^2}$ so ...
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1answer
44 views

What are some good integration problems where you can use some of the function convergence theorem of Lesbegue integrals?

I have learned about two major convergence theorem for the Lesbegue Measure: The monotone convergence theorem The dominated convergence theorem These are useful theorems for calculating integrals ...
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1answer
66 views

Show $\lim_{n \to \infty} n\cdot m(\{ x \in A | |f(x)| \geq n\}) = 0$

I have to show that $\lim_{n \to \infty} n\cdot m(\{ x \in A | |f(x)| \geq n\}) = 0$, for $(A, \textit{S}, m)$ a measure space and $f: A \rightarrow \mathbb{R}$ an integrable function. Let $A_n = ...
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2answers
124 views

fatou's lemma…about the inequality

I'm studying measure theory for the first time, and I just came across Fatou's Lemma. Why isn't it true that for any sequence of functions $\left\{ f_n \right\}$ in $L^+$ we always have that $\int ...
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2answers
67 views

About assumptions in the monotone convergence theorem

Why is the hypothesis that $\left\{f_n \right\}$ be an increasing sequence essential to the monotone convergence theorem? Could someone provide a nice, easy to understand counterexample if I were to ...
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1answer
72 views

Confusion regarding proof using Fatou's lemma

This is in reference to the book Problems in Mathematical Analysis III by Kaczor and Nowak. We are given that ${f_n}$ converges to $f$ on $R$. Suppose that $\lim_{n \to \infty} \int_R{f_n dm} - \int_R ...
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2answers
61 views

Integral construction on $L^1(0,a), a>0$.

I am working on the following problem: Let $a > 0$, $f \in L^1(0,a)$ and define $$ g(x) = \int_x^a f(t) t^{-1} dt, \quad 0 < x \leq a. $$ Show that $g \in L^1(0,a)$ and $\int_0^a ...
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1answer
300 views

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$.

Suppose $1\le p < r < q < \infty$. Prove that $L^p\cap L^q \subset L^r$. So suppose $f\in L^p\cap L^q$. Then both $\int |f|^p d\mu$ and $\int|f|^q d\mu$ exist. For each $x$ in the domain ...
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0answers
39 views

Integral of continuous function over probability distribution

Let $\mu_f$ denote the probability distribution of $f$ with $f(x)=10x-1$ for $x\in(0,1/2]$ and $f(x)=1$ for $x\in[1/2,1]$. If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function, what is ...
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2answers
48 views

Bernstein set in [0,2]

I have got a question to suggest an example of outer measure that is strictly not additive. I have thought about some special sets such as Bernstein set or Vitali set in an interval to suffice. ...
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1answer
47 views

Decreasing sequence of measurable functions

Suppose $f_1(x),f_2(x),\ldots:[0,1]\rightarrow\mathbb{R}$ are measurable functions such that $f_1(x)\geq f_2(x)\geq\ldots$. (infinite sequence) and $\lim_{n\rightarrow\infty}f_n(x)=0$. Is it true that ...
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1answer
61 views

Lebesgue integral equivalent statements

Suppose $\mu(\Omega)<+\infty$. Let $(f_n)_n\subset L^1(\Omega,\mu)$ such that $\int_{\Omega}|f_n|d\mu\leq K<\infty$ for all $n\geq1$. Prove this statements are equivalent: ...
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1answer
91 views

Necessary conditions for “convergence in mean” of a sequence of integrable functions.

Let $(X,\mathscr F,\mu)$ be a measure space (not necessarily finite) and $\{f_n\}_{n\in\mathbb N}$ and $f$ be nonnegative, real-valued and integrable functions on $X$ such that $f_n\to f$ almost ...
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1answer
73 views

Markov inequality limit

Let $f: \mathbf{R}^d \rightarrow [0,\infty]$ be a Lebesgue integrable function. Prove $$\lim_{\alpha \rightarrow \infty} \alpha m(\{x:f(x)>\alpha\})=0.$$ Hint: For $\epsilon>0$ take $g$ simple ...
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0answers
55 views

Uniqueness of a weak derivative

Let $f\in \mathrm{L}^2[a,b]$. As usually $f'$ is the so called weak derivative of $f$ if $\forall \phi \in C_c^{\infty}(a,b)$ $\int_a^b f'\phi dt=-\int_a^bf \phi' dt$. Is it reasonably to think that ...
0
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1answer
46 views

Representation of differentials in Polar Coordinates

We define polar coordinates in $\mathbb{R}^{n}$\ $\{ 0\}$ by $x=ry$, where $r=|x|>0$ and $y \in \partial B(0,1)$ is a point on the unit sphere. In the coordinates, Lebesgue measure has the ...
3
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2answers
78 views

Monotone convergence theorem to evaluate improper integral

My book says that the equality $$\int_{(0,1]}x^{-3/4}d\mu=\lim_{t\rightarrow 0^+}\int_{[t,1]}x^{-3/4}d\mu$$ follows from the monotone convergence theorem. Why is it so? I can't see how to apply ...
6
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1answer
63 views

Consequence of Cauchy Schwarz in $\mathscr{L}^2$?

If $f,g\in \mathscr{L}^2$, then $\|fg\|_1\leq\|f\|_2\|g\|_2$. My textbook says that this is a consequence of Cauchy-Schwarz inequality. How so? Cauchy-Schwarz says that $|\langle ...
1
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1answer
90 views

Integral of product of two measurable functions

I have to show that for $\psi$ and $\phi$ two positive and measurable functions: $$\int_X \phi\psi \, d\mu \le \sqrt{\int_X \phi^2 \, d\mu} \sqrt{\int_X \psi^2 \, d\mu}$$ I know that for $f,g$ two ...
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1answer
32 views

How do you call functions integrable over any compact subset of their domain?

I'm quite sure there is a name for such class of functions but can not remember or figure out what terms to search for. A simple and very practical example would be "periodic Lebesgue" functions: ...