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.

learn more… | top users | synonyms

3
votes
1answer
29 views

an argument that strengthen Lusin's theorem

Let $f$ be a measurable function on a subset $E$ of $\mathbb{R}^n$. Lusin's theorem states that for any $\epsilon>0$, there exists a measurable subset $F$ such that $F$ open in $E$, ...
0
votes
0answers
27 views

Let $f ∈ L_1((0,1)),$ and define $g : (0,1) → \mathbb{R}$ by $g(x)= \int^1_x \frac{f(t)}{t}dt$ [duplicate]

Let $f ∈ L_1((0,1)),$ and define $g : (0,1) → \mathbb{R}$ by $g(x)= \int^1_x \frac{f(t)}{t}dt$ Prove that $g ∈ L_1((0, 1)).$ Some help would be awesome. I tried doing this directly from definition ...
0
votes
1answer
24 views

Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ for all $M >0.$

Let $f$ be a measurable function on a measure space $(X,μ),$ where $μ$ is a finite measure. Suppose there are constants $K > 0$ and $p > 1$ such that $μ\{x∈X : |f(x)|>M\}< \frac{K}{M^p}$ ...
0
votes
0answers
34 views

Show that $lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} dx= \frac{\pi}{2}$. [duplicate]

I came across this qualifying exam problem and wasn't sure what to do. Using techniques of real analysis (as opposed to complex analysis) show that $lim_{R\rightarrow \infty}\int^R_0 \frac{\sin x}{x} ...
0
votes
1answer
74 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
1
vote
2answers
72 views

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions defined on a measure space $ (\Omega,\Sigma,\mu) $, where the index ...
0
votes
1answer
25 views

Prove that the set $A$ is measurable and find its Lebesgue measure.

Let $A ⊂ [0, 1] × [0, 1]$ be the set of points $(x, y)$ with decimal representations $x = 0.x_1x_2 ..., y = 0.y_1y_2 ...$ such that $x_ny_n = 5$ for all $n ∈ \mathbb{N}.$ Prove that the set $A$ is ...
1
vote
1answer
21 views

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$

Let $E⊂\{(x,y)|0≤x≤1, 0≤y≤x\}, E_x =\{y|(x,y)∈E\}, E^y =\{x|(x,y)∈E\}$ and assume that $m(E_x) ≥ x^3$ for any $x ∈ [0, 1].$ (a) Prove that there exists $y ∈ [0,1]$ such that $m(E^y) ≥ \frac{1}{4}.$ ...
1
vote
0answers
39 views

Prove that $\sup_n |T_n f(x)|<\infty,$ for a.a. $x$ for every $f \in L^1(R)$.

Denote $K(x)=\frac{1}{\sqrt{|x|}\cdot(1+x^2)}$ and $K_n(x)=nK(xn), n\in\mathbb{N}$. For $f\in L^1(R),$ define $T_nf(x)=\int_R K_n(x-y)f(y)\, dy.$ Prove or disprove that for every $f\in L^1(R), ...
1
vote
1answer
16 views

Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$

Let $(X,A,μ)$ be a $σ$-finite measure space with $μ(X) = ∞.$ Construct a function $F : X → \mathbb{R}$ such that $F ∈ L_p(μ)$ for all $p > 1,$ but $F \notin L_1(μ).$ I could easily do this if I ...
1
vote
3answers
108 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 ...
1
vote
0answers
41 views

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ [duplicate]

Let $f ∈ L_1([0,1])$ be a function such that $\int_E f(x)dx = 0$ for any measurable set $E ⊂ [0,1]$ of Lebesgue measure $0.99.$ Prove that $f = 0$ a.e. Not sure how to start this question. Any ...
0
votes
0answers
37 views

Suppose $f : [0,1] → R$ satisfies $f(x) − f(y) < x − y$ for all $x,y ∈ [0,1],x > y.$

Suppose $f : [0,1] → R$ satisfies $f(x) − f(y) < x − y$ for all $x,y ∈ [0,1],x > y.$ Show that $f′$ exists almost everywhere on $[0, 1]$ or give a counterexample. Not really sure how to go ...
1
vote
1answer
34 views

One dimensional integrals in Green's theorem

I am trying to understand Green's theorem, but the problem is I don't know what is the definition of the integrals in the theorem. This is the expression that one proves to hold with some assumption ...
0
votes
0answers
30 views

Prove that for any measurable set $A ⊆ \mathbb{R}$ $\int_A g_n dm → \int_A f dm.$ [duplicate]

Let $f, g_1, g_2 . . . ∈ L_1(\mathbb{R})$ be non-negative functions. Assume that $g_n → f$ a.e. and $\int_\mathbb{R} g_n dm = \int_{\mathbb{R}} f dm$. Prove that for any measurable set $A ⊆ ...
0
votes
2answers
36 views

$m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ Prove that $m(E) = 1.$

Let $E$ be a measurable subset of $[0, 1].$ Assume there is a constant $α > 0$ such that $m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ (Here $m(·)$ denotes Lebesgue measure.) Prove that $m(E) = 1.$ ...
1
vote
1answer
41 views

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Let $f$ and $g$ be integrable functions on a measure space $(X,Σ,μ).$ For each $t ∈ \mathbb{R},$ consider the sets $F_t =\{x∈X :f(x)>t\}, G_t =\{x∈X :g(x)>t\}.$ Prove that $\int|f − g| = ...
1
vote
0answers
20 views

Change of variable for Lebesegue Integral

Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds ...
0
votes
3answers
19 views

Square Integrable and Continuous

I have come across the notation $L^2(\Omega) \cap C(\Omega)$; while I believe understand the resulting behavior, I can't get my head around the machinery. When I think about $f \in C(\Omega)$ I am ...
4
votes
1answer
56 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
2
votes
1answer
65 views

Suppose that all the functions ${f_n},f$ are integrable. Is $lim_{n \rightarrow \infty} \int f_n(x)dx = \int f(x)dx?$

Let ${f_n(x)}$ be a sequence of continuous, strictly positive functions on $\mathbb{R}$ which converges uniformly to the function $f(x).$ Suppose that all the functions ${f_n},f$ are integrable. Is ...
6
votes
3answers
113 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
0
votes
0answers
31 views

Question about a theorem concerning the continuity of integral functions

If we have $$F(t):=\int_V f(t,x)dx$$ where $V$ is some measurable subset of $\mathbb R$ and $x\mapsto f(t,x)$ is a measurable function. Moreover let $F$ be defined for all $t\in U$ a open subset of ...
2
votes
0answers
25 views

Question on Integrability of a function.

Let $(f_j)_{j\in\mathbb{N}}$ be a sequence of real functions in $L^p(X,\mathcal{A},\mu)$, where $p\geq1$. If we know that ...
1
vote
2answers
67 views

Support of a positive measure

Let $\mu$ be Lebesgue measure on $\mathbb{R}$ , $\mathcal{M}$ be Lebesgue $\sigma-$ algebra and $f\in C_{c}\left(\mathbb{R}\right)$ (continuous with compact support). Suppose $f\geq0$ over ...
1
vote
1answer
60 views

Special case of Fatou's lemma

Hi everyone: Suppose that $D$ is a domain of $\mathbb{R}^{n}$ $(n \geq1)$, $Y$ a locally compact topological space, and $\mu$ a measure on $Y$. Let $f(x,y): D\times Y\rightarrow[0,+\infty)$ be ...
1
vote
3answers
68 views

Showing that $\frac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is lebesgue integrable

I'm having some real trouble with lebesgue integration this evening and help is very much appreciated. I'm trying to show that $f(x) = \dfrac{e^x + e^{-x}}{e^{2x} + e^{-2x}}$ is integrable over ...
0
votes
1answer
42 views

Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$

I need some help on the following real analysis past qual problem. I would appreciate some help. Let $E ⊂ [0,1]$ be a measurable set, $m(E) ≥ \frac{99}{100} .$ Prove that there exists $x ∈ [0,1]$ ...
0
votes
1answer
200 views

$\sum_{n = 1}^\infty f_n =f.$ Prove that $f′_n =f′$ a.e.

I am studying for a real analysis qualifying exam. Was hoping that there was a very slick proof for this? Thanks. Let $f_1, f_2, . . . , f : [0, 1] → \mathbb{R}$ be non-decreasing right-continuous ...
2
votes
1answer
28 views

Showing function is lebesgue integrable

I have a function $f(x) = \frac{\sin(\frac{1}{x})}{1+(\log(x))^2}$ and I am trying to find whether this is Lebesgue integrable on $[1,\infty)$. I'm really not sure where to start on this one. It ...
4
votes
3answers
176 views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
0
votes
0answers
42 views

Functions which can not be integrated via Riemann.

I am looking for some (possibly exotic) functions which can not be integrated via Riemann integration but can be integrated via Lebesgue. I am aware of the rational indicator function, i.e., $$ f(x) = ...
0
votes
2answers
23 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
1
vote
1answer
42 views

Looking for “explicit” integrals solvable using lebesgue integration theory

I am preparing for an exam in Measure and Integration Theory (Lebesgue Integration). As far as I know my professor prefers to ask students solving explicit integrals which can be solved using the main ...
2
votes
1answer
38 views

Reversing limits in Lebesgue integration

I know that reversing limits of Riemann integration is possible by putting minus sign. My question is that there is a similar result for Lebesgue integral as well. For example, $$ ...
2
votes
0answers
43 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
2
votes
2answers
91 views

If $f$ and $f'$ are integrable, then $f'$ has integral $0$

I have the problem Prove that if $f$ and $f'$ are Lebesgue integrable over $\mathbb{R}$, then $\int_\mathbb{R}f' = 0$, where $f'$ is defined everywhere. Honestly, not sure where to start. I had ...
2
votes
1answer
66 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
0
votes
1answer
33 views

Would like to compute the limit of some integral

I was working on a exercise where the goal was to compute the following limit, $$\lim_{n\rightarrow\infty}\int_{\mathbb R}e^{-|x|n}e^{-\frac{x^2}{2}}dx$$ and some tutor of mine claimed that the limit ...
0
votes
1answer
34 views

Monotone property of Lebesgue Intergral

Let $(X,\mathcal{A},\mu)$ be a a measure space, and $f$ and $g$ two measurable functions. Now if $f$ and $g$ are nonnegative and $f\leq g$, it can be easily seen that $\int f\,d\mu\leq \int g\,d\mu$, ...
2
votes
1answer
51 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
1
vote
0answers
67 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
votes
1answer
36 views

counterexample of Riemann-Lebesgue lemma for non-Borel functions

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a Borel measurable function. Then $$ \lim_{\lambda\to\infty}\int_{\mathbb{R}}f(x)e^{i\lambda x}d\mu(x)=0. $$ I obtain this result by showing that it is ...
2
votes
1answer
76 views

A question on almost every where

Let $(X,\mathcal{A},\mu)$ be a measure space and $u$ some measurable function. If $v$ is a function and $u=v$ a.e.; when is true to say that $v$ is measurable? Also let P be some property, and $f$ ...
0
votes
1answer
46 views

Condition for integrability on finite measure space

Let $(X,\mathcal{F},\mu)$ be a finite measure space. If $f:X\rightarrow \mathbb{R}$ is a measurable real function, show that, $f\in L^1(\mu)$ iff $\sum\limits_{n=0}^{\infty}\mu(\{f\geq ...
0
votes
1answer
17 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
1
vote
1answer
49 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
1
vote
2answers
57 views

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too?

If $X,Y \subset \mathbb{R}$ are measure zero sets, how can I show that $X \times Y \subset \mathbb{R^2}$ is a measure zero set too? My outline is the following: Since $X,Y$ is a measure zero set, ...
2
votes
1answer
48 views

Integral equation $f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy$

I'm trying to solve the following equation $$f(x) = e^{-cx} + \lambda\int_0^xce^{-cy}f(x-y)dy,\quad x>0 $$ where $c$ and $\lambda$ are constants and $f$ is a continuous bounded function on ...
2
votes
1answer
52 views

Calculating a Lebesgue integral

Calculate the Lebesgue integral of, $$\lim_{n\to\infty}\int_{[0,1]}\frac{n\sqrt{x}}{1+n^2x^2}$$ I know I should use the Lebesgue dominated convergence theorem but what should be the dominating ...