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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13 views

Approach a length by a BV norm

Let $\Omega$ be a smooth bounded open domain in $\mathbb R^d$. Let $g: \overline{\Omega}\to \mathbb R^+$ defined by $g(x)=f(x)$ if $x\in \Omega$ and $g(x)=h(x)$ if $x\in \partial \Omega$, where ...
2
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0answers
18 views

Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

What I've done: $$ \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = ...
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0answers
22 views

Evaluate $\int_{0}^{\infty} \frac{\sinh bx}{\sinh ax} dx $

I need to evaluate the following integral $$\int_{0}^{\infty} \frac{\sinh bx}{\sinh ax} dx \space \space \space , \space \space 0<b<a$$ Here is my attempt - I can write $\sinh ax $ as ...
3
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2answers
50 views

When do we have the formula $f(t)=e^{\lambda t}f(0)+\int_0^te^{\lambda (t-s)}g(s)ds$?

Let $g:\mathbb{R}\to \mathbb{R}$ be a continuous function. Consider the following integral equation $$f(t)=f(0)+\int_0^t\lambda f(s)ds+\int_0^tg(s)ds. \tag{1}$$ Since $g$ is continuous, Thus the ...
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1answer
18 views

How to find the inverse Fourier transfmation of exp(-sk)/k.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of exp(-sk) which is $$ \frac{\sqrt2}{\sqrt pi}\frac{x}{x^2+ s^2}$$ .After ...
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1answer
25 views

Monotonic increasing and convergence in measure

If for each $n\in\mathbb{N}$, $f_n$ is monotonic increasing on [0,1] and $f_n\rightarrow f$ in measure, then $f_n\rightarrow f$ at every x at which f is continuous. I'm not sure whether this is right ...
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16 views

Relationship between Convergence in mean, convergence in measure and a.e. convergence

What is the relationship between convergence in mean under 1-norm (http://mathworld.wolfram.com/ConvergenceinMean.html), convergence in measure and a.e. convergence? I have shown that convergence in ...
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34 views

Lebesgue measurable integration, density

Let $\mathbb{T}$ be the unit circle and $\lambda$ be the Lebesgue measure on $\mathbb{T}$. Let $A_n := e^{2\pi i[1/2^{2n},1/2^{2n+1}]}$, $n\ge 1$. Define a function $f$ on the set of all the Lebesgue ...
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2answers
18 views

Show $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$

Suppose X and Y are integrable random variables on the measure space $(\Omega,\mathcal F, P)$. Im trying to show that $E(Y)-E(X) = \int_{\mathbb R} P[X<t\le Y] - P[Y< t \le X] dt$ but I got ...
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1answer
68 views

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$.

Compute the Lebesgue integral $\int_0^{\infty} \frac{x}{e^x -1}dx$. I think I need to use the Dominated Convergence Theorem or the Beppo Levi Theorem to show this, but I don't really know what I ...
2
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1answer
54 views

$f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$?

Suppose that $f: \mathbb{R} \to \mathbb{R}$ is Lebesgue integrable. Does it follow that $\lim_{x\to \infty} f(x)=0$? What if $f$ is continuous on $\mathbb{R}$? I think the first question is false but ...
2
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1answer
62 views

Application of Fubini-Tonelli's Theorem on function $\frac{2}{\pi}e^{-ax}\cos(x\cos{\theta})$

The question asks me to prove that $$\int_0^\infty J(x)e^{-ax}dx=\frac{1}{\sqrt{1+a^2}},$$ where $a>0$ and $J(x)=\frac{2}{\pi}\int_0^{\pi/2}\cos(x\cos{\theta})d\theta.$ I started off by ...
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1answer
22 views

Function of bounded variation and integration

Let f belong to $C[a,b]$. Show that there is a function g that is of bounded variation on [a,b] for which $\int_a^bfdg=||f||_{max}$ and TV(f)=1. This problem appears on page 162 of Royden's Real ...
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2answers
45 views

Markov Inequality proof (measure theory)

I am trying to prove Markov's Inequality in measure theory as: Let $g:\mathbb{R}\rightarrow \mathbb{R}$ be a non-negative function which satisfies $g(x)>0$ se $x>0$, and not descendant in ...
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0answers
64 views

Proof that $f = 0$ almost everywhere

My question is about the proof of parts (a) and (b) of Theorem 1.39 on page 30 of Rudin's "Real and Complex Analysis." 1.39 Theorem. DIFFICULTY # 1: (a) Suppose that $f : X \to [0, \infty]$ is ...
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0answers
21 views

Equivalent conditions for weak $L^p$ spaces for $p\leq 1$

I have difficulty doing the following exercise from Tao's real analysis book: Let $X$ be $\sigma$-finite measure space and $0<p\leq 1$. Then show that the following are equivalent: $f$ is in ...
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1answer
63 views

Show that there exists a continuous function $f$ such that $\int |\chi_A-f| d\lambda\lt \epsilon$

Let $\lambda=l^*$ denote Lebesgue measure on $\Bbb R$, and let $A$ be a Lebesgue measurable set with $\lambda(A)\lt +\infty$. Show that if $\epsilon \gt0$, there exists an open set which is the union ...
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2answers
47 views

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$

Let $f:\Bbb R \rightarrow \Bbb R$ be a Lebesgue measurable function that is in $L^2$. Show $F(x)=\int_0^x f(t)dt$ satisfies $|F(x)-F(y)|\leq C|x-y|^\frac 12$. Here's what I have so far. $f\in L^2 ...
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3answers
42 views

Prove that a function is $L^p(\mathbb{R})$

There is a specific criterion for proving that a function $f \in L^p(\mathbb{R})$ as well as proving it by definition ? Furthermore, is correct to imply that: If $|\ f|^{\ p}$ is continuous in ...
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18 views

Existence of a locally essentially unbounded integrable function

Does there exist an integrable function $f\colon [0,1]\to \mathbb{R}_+$ such that for every $0\leq a < b\leq 1$ we have $\| \chi_{(a,b)} f\|_\infty = + \infty$?
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1answer
34 views

half-closed intervals and Lebesgue measures

I am reading Bartle's book. define $$K=\{ a \in \mathbb{Q}\,|\, 0 < a \le 1\}$$ and define $A$ by the family of all finite unions of half-closed intervals in the form of $$\{a \in K\, |\, x ...
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3answers
43 views

Show that continuous functions on $[0,1]$ satisfy this property

If $f \in C[0,1]$ prove that $$ \lim_{n \to \infty} n\int_0^1e^{-nx}f(x)dx $$ exists and find the limit. I can show that $|g_n|$ are bounded by $M=\max(f)$. After some test functions I suspect that ...
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2answers
36 views

Lebesgue Integrability of $\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$

Given $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(0)=0$ and $f(x)=\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$ for $x\in \mathbb{R}-\{0\}$, can someone please give me a rigorous proof ...
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21 views

what is the relation between X and ω

From the definition of random variable: In the special case of probability space (Ω, F, P), we use the phrase random variable (RV) to mean a measurable function, that is, X : Ω → R is a random ...
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1answer
26 views

Given convergence of integrand and integral, show convergence of integral over arbitrary measurable set

All measures are Lebesgue. $\forall n \in \mathbb{N}$, let $f_n: \mathbb{R} \rightarrow [0, \infty]$ be measurable and almost everywhere $f_n \rightarrow f$; moreover, suppose that $\int f_n dλ ...
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0answers
30 views

Show that $\int\left\lvert f_n\right\rvert\,d\lambda\to\int\left\lvert f\right\rvert\,d\lambda\implies\int\left\lvert f_n-f\right\rvert\,d\lambda\to0$

Let $\,f, f_n $ be Lebesgue integrable functions mapping reals to extended reals such that, almost everywhere, $\,f_n \to f $. Show that ...
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1answer
31 views

Results on “subtraction” of measures and outer measures?

Most results I have seen involves addition of measures For example, let $m^*$ and $m$ be Lebesgue outer measure and Lebesgue measure respectively, then given $A = \bigcup\limits_{n = 1}^\infty E_n, ...
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0answers
24 views

Limit of integrals is zero

Let $\lambda$ be a lebesgue integral on $[0,1)$. Define the intervals $I_{n,i}=\left(\frac{2i}{2n}, \frac{2i+1}{2n}\right)$ and $J_{n,i}=\left(\frac{2i+1}{2n}, \frac{2i+2}{2n}\right)$ for $0\leq i\leq ...
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1answer
19 views

Prove $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k 0$.

Let $\mathcal{Q_k}=[-k,k]^n\subset \mathbb{R^n}$ for all $k\in\mathbb{N}$, the n-dimensional cubes, and $f$ any integrable (lebesgue) function. Prove that $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k ...
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4 views

Prove that $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k 0$.

Let $\mathcal{Q_k}=[-k,k]^n\subset \mathbb{R^n}$ for all $k\in\mathbb{N}$, the n-dimensional cubes, and $f$ any integrable (lebesgue) function. Prove that $\int_{\mathcal{Q_k}^c} \vert f \vert \to_k ...
0
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1answer
30 views

Decreasing sequence of non-negative Lebesgue measurable functions and MCT

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem: Suppose that $f$ and $f_n$ are nonnegative measurable ...
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0answers
12 views

Point-free notation for (not direct) sums of functions over products of spaces.

I am writing a paper, and there are lots of expressions containing integrals of form: $$\int_{X \times Y} \phi(x) + \psi(y) d\alpha(x,y) $$ where $\phi$,$\psi$ are abstract functions and $\alpha$ is ...
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3answers
34 views

Verification of a proof in Measure Theory

Let $m$ be the Lebesgue measure on $\Bbb R$ and $f:\Bbb R\to [0,\infty)$ be a Lebesgue integrable function. Show that $\exists $ a measurable set $E\subset [0,\infty)$ such that $m(E)\neq m(f^{-1}(E)$ ...
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1answer
50 views

Uniform continuity with integral being finite

Let $f$ be a real valued uniformly continuous function on $\mathbb{R}$ that is lebesgue integrable. Show that $\lim_{|x|\rightarrow \infty}f(x)=0$. Suppose that ...
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1answer
24 views

$\lVert f \rVert_{\Phi} < \infty$ for every $f$ measurable and satisfying a certain condition

Let $\Phi : [0, \infty) \rightarrow [0, \infty)$ a convex, strictly incrasing function, with $\Phi(0)=0$. Let $L^{\Phi}(0,1)=\{f:(0,1)\rightarrow\mathbb{R} \text{ measurable}:\int_0^1\Phi\left( ...
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1answer
70 views

A problem related to Lebesgue integration.

I have following two problems: Suppose $$\int_E f \, dx = 0 $$ where $ f: R \to R$ is a measurable function that is strictly positive. Show that $E$ must be a null set. Next Suppose that $E$ is a ...
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1answer
39 views

lebesgue and riemann integrals are the same for continuous functions on $[a,b]$

I have a proof in front of me which goes as follows, firstly assuming that the function $f \geq 0$ on $[a,b]$. We get a partition $a = x_0 < x_1 <....<x_n = b$ with $x_i - x_{i-1} = ...
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1answer
21 views

Multiple Lebesgue integrals: counting measure

I am completing an exercise on multiple Lebesgue integrals. The problem is as follows: Let $X=Y=\Bbb{N}$ and $\mathcal{A}=\mathcal{B}=\mathcal{P}\Bbb{N}$ with counting measures $\mu$ and $\nu$ on ...
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1answer
42 views

Show that $\int_{\mathbb{R}}f = \lim_{n \to \infty} \int_{\mathbb{R}}f_n$ given specific assumptions

I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem: If $f, f_n: \mathbb{R} \to [0, +\infty)$ measurable, $f_n ...
2
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1answer
37 views

Showing that the double integral of $e^{-xy}$ exists in $\{(x,y):0<x<y<x+x^2\}$

I want to show that $f(x,y)=e^{-xy}$ is Lebesgue integrable in the region $E=\{(x,y):0<x<y<x+x^2\}$ using Fubini's theorem. I thought I could rewrite the function as ...
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1answer
37 views

Two questions on Lebesgue integration and application of reverse triangle inequality

I'm learning about measure theory (specifically Lebesgue integration) and need help with the following problem: Let $f_n, f \in L^1$ and $\int_{\mathbb{R}}\left|f_n-f\right| \rightarrow0$. Prove ...
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1answer
25 views

$\int_{\bigcup_{n=1}^{\infty}E_n}f=\sum_{n=1}^{\infty}\int_{E_n}f$ given $f$ positive and measurable

I'm learning about measure theory (specifically Lebesgue intregation) and need help with the following problem: Let $f:\mathbb{R}\rightarrow[0,+\infty)$ be measurable and let $\{E_n\}$ be a ...
0
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0answers
39 views

Find the $\lim\limits_{c \to \infty} \int_{-\infty}^{+\infty} |g(x) - g(x+c)|dx$

Find the $$\lim_{c \to \infty} \int_{-\infty}^{\infty} |g(x) - g(x+c)|dx$$ where $g$ is integrable. I know already that if $g$ is integrable, than the integral of $g(x)$ and the integral of ...
1
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1answer
63 views

Lebesgue and Riemann integrals two proofs

1. Let X be a finite closed interval [a,b] in R, let X be the collection of Borel sets in X and let λ be a Lebesgue measure on X. If f is a nonnegative function on X, show that ∫ fdu =∫a->b f(x)dx I ...
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1answer
36 views

how to prove isometric vector space isomorphism

Let $(L^1)^*$ be the dual space to $L^1$, or bounded linear functional over $L^1$, i.e., $f:L^1\to \mathbb{R}$, $f(cx+y)=cf(x)+f(y)$, $|f(x)| \leq M|x|^1$ Define the norm for $f \in (L^1)^∗$ by ...
2
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0answers
30 views

Dominated convergence theorem vs continuity

Let $\{f_n\}$ be a sequence of functions in $L^2(0,1)$ such that $\lim_n f_n = f$ pointwise and $\vert f_n(x) \vert \leq g(x)$ for some integrable function $g$. By the dominated convergence theorem it ...
7
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2answers
82 views

Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$

Let $f : \mathbb{R} \to [ 0, \infty)$ be a measurable function. If $\int_{- \infty}^{\infty} f(x) dx =1$. Then I want show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$ Any help ...
1
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1answer
41 views

Show that $\mathcal{B}(X \times Y) \subseteq \mathcal{B}(X) \times \mathcal{B}(Y)$

Let $X$ and $Y$ be two locally compact Hausdorff spaces i. Show that each $f \in C_c(X \times Y)$ is a limit of sums of the form $$\sum\limits_{i=1}^n \varphi_i(x) \psi_i(y)$$ where $\varphi_i \in ...
2
votes
1answer
51 views

Exercise #9 in chapter 11 of Rudin's Principles of Mathematical Analysis.

Suppose $f$ is Lebesgue integrable on $[a,b]$. Let $F(x)$=$\int_{a}^x fdt$. Then prove that $F$ is continuous on $[a,b]$. I know that $F$ is continuous almost everywhere, because $F'(x)=f(x)$ ...
0
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0answers
17 views

Monotonicity property almost everywhere

I'm studying for a qualifying exam and having difficulty showing the following: Let $f\in L^1_{loc}(\mathbb{R})$ be a real-valued locally integrable function. Suppose that for each positive integer, ...