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
24 views

Proving that a function is L1

Suppose $f \in L^1([0,b])$ and $g(x)=\int_x^b{\frac{f(t)}{t}dt}$ , prove that $g\in L^1([0,b])$ and $\int_{0}^{b} g(x) dx = \int_{0}^{b} f(t) dt$. Assume we are not allowed to use integration by ...
2
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0answers
39 views

Averages of integral and$ L^p$ space problem

Let $f: \mathbb R \to \mathbb R$ be an integrable function, for each $h>0$ let $$f_h(t)=\dfrac{1}{h}\int_{t-\frac{h}{2}}^{t+\frac{h}{2}}f(x)dx$$ Suppose $f \in L^P$, prove the following (1) $f_h ...
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2answers
57 views

Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$.

Assume $E$ has finite measure and $1 \leq p < \infty$. Suppose $\{ f_n\}$ is a sequence of measurable functions that converges pointwise a.e. on $E$ to $f$. For $1 \leq p < \infty$, show that ...
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0answers
45 views

$f$ locally bounded, nonnegative, and measurable function integrable iff series $\int_{n=1}^{\infty}a_{n}$ converges absolutely

Suppose $f$ is a locally bounded, nonnegative, and measurable function on $[1,\infty)$ and define $\displaystyle \int_{n}^{n+1}f$, $\,\,\forall n \in \mathbb{N}$. Then, is it true that $f$ is ...
1
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1answer
34 views

Lebesgue integral of non-negative

Assume that $f: [0,1] \rightarrow [0,\infty)$ is a Lebesgue measureable function such that $f(x) > 0$ for a.e $x.$. Show that for every $\epsilon >0$ there is $\delta >0$ such that for every ...
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0answers
19 views

Calculate Norm Operator

I'm trying to solve this exercice: Let $\omega(y)=y^{-4}$ and $L^{1}(\mathbb{R},\omega)$ the space of measurable functions $g:\mathbb{R}\rightarrow\mathbb{R}$ so that $g\omega$ is Lebesgue ...
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4answers
56 views

Show that the function is not Lebesgue Integrable

Related to this question: How do I show that $f(x)=\frac{(−1)^{n}}{n}$ for every $n⩽x<n+$ and $n\geq 1$ is not Lebesgue Integrable? I'm extremely confused by the notation. What I need to show is ...
1
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1answer
23 views

Show that measure of symmetric difference $= \int_{E}\vert \chi_{A} - \chi_{B}\vert$

I have just proven that the Nikodym (pseudo)metric $\rho(A,B)$ is a pseudometric (i.e., satisfies all the metric axioms, except that $\rho(A,B)$ can be $0$ even if $A \neq B$), and now I need to show ...
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0answers
18 views

Unbounded Case of $\int_{[0,1]}x^{\alpha}$

For a number $\alpha \in \mathbb{R}$, define $f(x) = x^{\alpha}$ for $0 < x \leq 1$ and $f(0) = 0$. I am tasked with computing $\int_{[0,1]} f$ in both the bounded and unbounded cases. I assume ...
2
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0answers
46 views

Change of variable for integration with respect to Haar measure

I know how to estimate the integral \begin{gather} \int f(Ub)\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] \end{gather} where $f:S^n(\mathbb{R})\to \mathbb{R}$ ...
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2answers
37 views

The convolution of an integrable function with a $p$-integrable function is integrable

Let $\Sigma$ denote the set of Lebesgue-measurable subsets of $\mathbb{R}$, and $m$ the Lebesgue measure on $\mathbb{R}$. Let $1<p\leq \infty$, $f\in L^1(\mathbb{R},\Sigma,m)$, and $g\in ...
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0answers
23 views

Simple function with same support as a function with finite support

I am looking at a solution to the first part of problem 21 in Section 4.3 of Royden's Real Analysis. The part of the problem I am interested in states as follows: Let the function $f$be ...
-2
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1answer
29 views

Measure of boundary in $\mathbb R^n$

I saw that the measure of the boundary of a regular open set in $\mathbb{R}^n$ is zero, so how can we talk about the integral on this boundary (for me it must be equal to zero always )? I saw that in ...
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1answer
36 views

Prove that if E is positive lebesgue measurabel set, then E − E and E + E contain non-empty open sets.

let E + E = {x + y : x, y ∈ E}, and define E − E similarly. Show that if E is a measurable subset of R of positive Lebesgue measure then E − E and E + E contain non-empty open sets. I have seen the ...
1
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1answer
38 views

Fredholm integral operator

Let $T(f)(x):=\int_{\mathbb{R}^d} k(x,y)f(y) dy$ where $k_1(x)= \int_{\mathbb{R}^d} |k(x,y)|dy$ and $k_2(y)= \int_{\mathbb{R}^d} |k(x,y)|dx$ are two $L^{\infty}$ functions. Then I want to show that ...
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0answers
28 views

Can we use a series of properties to determine integral operator $f \to \int_0^1 f d\mu $

Question: Suppose there exists an operator $I: C^{\infty}(0,1) \to \mathbb R$ satisfying the following properties: (1) $I (\chi_{(0,1)})=1$ ; (2) $I(kf)=kI(f)$, where $k\in \mathbb R$ and $f\in ...
5
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3answers
169 views

If $\int_A f\,dm = 0$ for all $A$ having some fixed measure $C$, then $f = 0$ almost everywhere

Let $ f \in L^1[0,1]$. Assume that there is a constant C, with $0 < C < 1$, such that for every measurable set $A \subset [0,1] $ with $m(A)=C$, we have $ \int_{A} f dm = 0 $. Prove that ...
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0answers
55 views

Problems with Proof of Jensen's Inequality (Durrett's “Probability Theory and Examples”)

I have some questions concerning the proof of the Jensen's Inequality I found in Durrett's "Probability Theory and Examples" [pp.23-24]. In the following there is the proof, with the questions I have ...
2
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1answer
29 views

Bounded variation and Integration

This is an exercise in section 6.3 of Royden & Fitzpatrick's Real Analysis. Consider the function $$f(x)= \begin{cases} x^a\sin\left(\frac{1}{x^b}\right) & \text{if }0 < x \leq 1 \\ 0 ...
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0answers
18 views

Difference of increasing functions differentiable a.e.

I'm working through Royden & Fitzpatrick's Real Analysis, in the beginning of section 6.3 it reads and I quote: "Lebesgue's theorem tells us that a monotone function on an open interval is ...
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2answers
65 views

Use the definition of the integral to prove that if $A \subset E$, where $E$ is measurable, then $\int_A f = \int_E f \chi_{A}$

I want to prove: If $A \subset E$, where $E$ is measurable, then $\int_A f = \int_E f \chi_{A}$, where $f$ is a bounded measurable function and $m(E) < \infty$ Solution: Let $f: E \to ...
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0answers
31 views

Expectation and the Survival Function: Measure Theory

I have an example from class notes that I do not understand and would appreciate some clarification. Particularly, I haven't found a direct explanation online or in my text with regards to the limits ...
1
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1answer
25 views

Example of a Lebesgue integrable function under certain conditions

I need to find a sequence $(f_k)$ of Lebesgue integrable functions such that $f_k \to 0$ almost everywhere but $\lim _{k\to \infty} \int |f_k| \ne 0$. Here is an example that I thought: $f_k(x) = ...
0
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0answers
34 views

Antiderivative $f(x)=\int_{a}^{x} g(x)$ is differentiable almost everywhere on (a,b)

An exercise in Royden & Fitzpatrick asks to show that if g is integrable on [a,b] and we define the antiderivative of g as: $f(x)=\int_{a}^{x} g(x)$ for all $x\in[a,b]$, then f is differentiable ...
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3answers
68 views

prove that $ \int_E f(x) dm \ge \delta$ whenever $m(E) \ge \epsilon$

Assume that $f: [0,1] \to [0, \infty) $ is a Lebesgue measurable function such that $ f(x)\gt 0 $ a.e x. Show that for every $ \epsilon \gt 0 $ there is $\delta \gt 0$ such that for every lebesgue ...
0
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1answer
35 views

If given conditions are satisfied, then prove that $f$ is absolutely continuous on any interval $[a,b]$

Assume that $ f: R \to R $ is a non-decreasing function with $ \int_R f' dm =1, $ $ \lim_{x \to-\infty} f(x) =0 $ , $ \lim_{x \to\infty}f(x)=1 $. Then Prove that $f$ is absolutely continuous on any ...
1
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1answer
34 views

$ \lVert {\bf f} \rVert_{p} \leqslant (m(E))^{1-1/p} \lVert {\bf f} \rVert_{\infty} $ for any $ f \in L_{\infty}(E) $

If $m(E)$ is finite and ${\bf f}\in L_\infty(E)$ then for any $p\geqslant 1$, $$ \lVert {\bf f} \rVert_{p} \leqslant (m(E))^{1-1/p} \lVert {\bf f} \rVert_{\infty} .$$ I tried to apply Hölder's ...
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3answers
35 views

Show that there is no Lebesgue integrable function $g$ satisfying $n\chi_{(0,\frac1n]}\leqslant g$ for all $n$.

Show that there is no Lebesgue integrable function $g$ satisfying $n\chi_{(0,\frac1n]}\leqslant g$ for all $n$, where $\chi$ is the characteristic function. I tried to reason by contradiction. ...
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0answers
21 views

Convergence in norm

If {$f_k $} is a sequence of Lebesgue integrable functions, then {$f_k$} is said to "converge in norm" to an integrable function $f$ if $\int | f_k - f | $ converges to zero . Can someone explain to ...
1
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2answers
46 views

Is the function $f(x,y) = \frac{x^2 - y^2}{(x^2+y^2)^2}$ lebesgue integrable in $[0,1]\times[0,1]$?

So, there's something that I can use to show that $f(x,y) = \frac{x^2 - y^2}{(x^2+y^2)^2}$ is or is not Lebesgue integrable in $[0,1]\times[0,1]$?
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1answer
19 views

Lebesgue integrability of step functions

A step function can be defined to be a linear combination of a sequence of brick functions. My question is - Are step functions always Lebesgue integrable ?
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0answers
28 views

Show that there is a subsequence of $F_n$ converging to zero almost everywhere.

If F is a positive $L^1$ function on R. Define $F_n = F(x+n)$. Show that there is a subsequence of $F_n$ converging to zero almost everywhere. If I can show that $F_n$ converges to 0 in measure or ...
3
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5answers
112 views

find the limit: $\lim_{n\to\infty}\int_{0}^{\infty} \frac{\sqrt x}{1+ x^{2n}} dx$

Calculate the following limit $$\lim_{n\to\infty}\int_{0}^{\infty} \frac{\sqrt x}{1+ x^{2n}} dx$$ I tried to apply dominated convergence theorem but I could not find the dominating function. even I ...
2
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4answers
94 views

$e^{-x^{\alpha}}$ is Lebesgue-integrable on $[0,\infty)$ for $\alpha>0$

Prove that for $\alpha>0$, the function $e^{-x^{\alpha}}$ is $\mu$-integrable on $[0,\infty)$, where $\mu$ is the one-dimensional Lesbesgue measure. Obviously we want to make an upper bound ...
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1answer
20 views

Integrand of a definite integral

If we have an integral $\int_A f(x)dx$, or what I am more specifically interested in, $\int_A fd\mu$ where $\mu$ is a measure, is the integrand $f$ or is it $f$ on the domain $A$, where A is some set ...
3
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2answers
61 views

If the measure of a set is zero, the integral over the set is zero and the function is zero a.e?

Note: This wall of text is overkill and probably not too representative of the question in the title. I recommend you simply skip to edit 4, and then see the answers below. Let us consider the ...
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2answers
44 views

Prove that $\mid f(x)\mid \le 1 $ for almost all x.

Suppose $f$ is a real valued $L^1$ function on $\mathbb R$ such that for all measurable sets $E$, we have $$\left| \int_E f(x)\ \mathsf d m \right| \le m(E) $$ where $m$ is Lebesgue measure. ...
0
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1answer
16 views

in the Fourier inverse formula for a distribution function, the integral is Lebesgue-integrable?

To keep things simple, let $X$ be a random variable, $F$ its distribution function, and $\phi$ its characteristic function. If $0$ is a continuity point of $F$, then $$ ...
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2answers
38 views

Lebesgue integral problem relating Dirac measure

Let $\delta_a$ be the Dirac measure and $\mu_n = \frac{1}{n} \sum_{i=1}^n \delta_{i/n}.$ Show that $\lim_{n\to \infty} \int_0^1 f(x) \mu_n(dx) = \int_0^1 f(x) dx$ and if this is true, does it imply ...
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0answers
11 views

Integration of subdifferential.

For example, for the function $$ f(x)=|x|,\quad\text{$-1\le x\le1$,} $$ the subdifferential $D^{+}f(x)$ is $$ D^{+}f(x)=\begin{cases} -1\quad&\text{for $x\in[-1,0)$,}\\ [-1,1]\quad&\text{at ...
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0answers
39 views

Show that $f$ is a measurable function if $f$ is bounded

Let $m(E)=0$. Show that if $f$ is a bounded function on E, then $f$ is measurable, and the Lebesgue integral is zero. I was thinking of using this theorem that state that $\int f =0 $ iff $f=0$ that ...
0
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1answer
84 views

Use definition of Lebesgue integral for nonnegative functions to show $\int_{A}f=\int_{E}\chi_{A}f$.

I am trying to show the following: Let $f$ be a measurable function on a set $E$, and let $A$ be a measurable subset of $E$. Then, using the definition of the Lebesgue Integral of nonnegative ...
1
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0answers
26 views

Definition of Simple Functions

Recently I came across an alternative definition for simple functions, namely that a function $f: X \rightarrow \mathbb{R}$ on a measure space $(X, \mathcal{B})$ is simple if it only takes on a finite ...
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1answer
36 views

Measurable function and difference set

I don't understand this task and how to solve it: Let $\lambda$ denote the Lebesgue measure on ($\mathbb{R}$,$\mathfrak{B}(\mathbb{R})$). Recall that for any $E \in \mathfrak{B}(\mathbb{R})$ with ...
2
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1answer
28 views

A stronger form of the weak $(1,1)$ inequality for the Hardy-Littlewood maximal function

I am trying to show that for $f \in L^1(\mathbb R^d)$, if $f^*(x)$ is the Hardy Littlewood Maximal function, then the following inequality is satisfied:$$|\{x : f^*(x)> \alpha\}|\leq ...
0
votes
1answer
142 views

Is $\sup_{\| f \| \leq 1}{\left| \int f d\mu \right|} = \sup_{\| f \| \leq 1}{\{ |\mu(f)| \}}?$

The answer given by t.b. mentioned the following One of the most convenient way of writing a total variation norm is $$\| \mu \| = \sup_{\| f \| \leq 1}{\left| \int f d\mu \right|}$$ In ...
2
votes
2answers
53 views

Why this doesn't contradict Monotone and Dominated Convergence Theorem? [closed]

$$\lim_{n \to \infty} \int_0^\infty f_n(x)dx \ne \int_0^\infty \lim_{n \to \infty} f_n(x)dx$$ where : $f_n=ne^{-nx}$ for all $x \in [0,\infty)$ $n \in \mathbb{N}$ Can somebody help me with ...
1
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1answer
44 views

Question about inequality relating infinity norm and Lebesgue integral

I have a question about the following question: Let $f:\mathbb{R} \to \mathbb{R}$ be a measurable function. We know there exists a constant $K$ such that for every bounded continuous function ...
0
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0answers
24 views

Fourier transform isometry

I want to show that for a sufficiently fast decaying function we have $$\pi \int_{\mathbb{R}^3} \frac{f(x)}{|x|} dx = \int_{\mathbb{R}^3} \frac{\widehat{f(x)}}{|x|^2} dx. $$ Does anybody see where ...
2
votes
1answer
34 views

Attempting to use dominated convergence theorem

I am trying to prove the following: Let $(X,\Sigma, \mu)$ be a measure space and let $f:X\rightarrow [0,\infty)$ be an integrable function. Then for any real number $\alpha>1$, $$\int_X n ...