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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16 views

Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain ...
3
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1answer
46 views

Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Then $\int_{(0, 1)}\sum_{n=1}^{\infty}f_n \neq \sum_{n=1}^{\infty}\int_{(0, 1)}f_n.$

I'm learning about measure theory, specifically Lebesgue integration, and need help to understand the solution to the following problem: Let $f_n(x) = nx^{n-1}-(n+1)x^n$, $x\in (0, 1)$. Show that ...
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2answers
62 views

Prove the monotone convergence theorem for sequences of Lebesgue-integrable functions

I'm trying to prove the monotone convergence theorem for $L^1$ functions: Suppose $(f_n)$ is a sequence of $L^1$-functions (i.e Lebesgue-integrable functions) over a measure space $(X,\sigma (X), ...
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25 views

conditions on integrable function with counting measure

Let $P(N)$ be the power set of $N$ and $u$ be the counting measure on $N$. (a) Prove or disprove the measure space $(N, P(N), u)$ is complete? (b) Given function $g: N\rightarrow R$. Show that $g$ ...
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1answer
28 views

Integration obeys countable subadditivity?

Does Lebesgue integration have the property of countable subadditivity: 'if $f$ is integrable on $E$ and $E = \bigcup_{i=1}^{\infty} E_n$ then $\int_E f \le \sum_{i=1}^{\infty} \int_{E_n} f$'? You ...
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0answers
62 views

Why does this integral not contradict Fubini's Theorem?

I have the integral: $$\int^{1}_{0}\int^{\infty}_{1} (e^{-xy}-2e^{-2xy}) \,\text{d}y~\text{d}x$$, and I know that the order of integration cannot be interchanged, but why does this not contradict ...
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0answers
32 views

When $F(t)=\int_0^tf(s)ds$ is differentiable everywhere?

Let $f:\mathbb{R}\to \mathbb{R}$ be a function that is continuous almost everywhere. 1) Is the function $F(t)=\int_0^tf(s)ds$ differentiable everywhere ? 2) What is the "weakest" condition on $f$ ...
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1answer
25 views

Given $\lim\limits_{x\to\infty} f(x) = r$, show $\lim\limits_k\int_{[0,a]}f(kx) = ar$

Show $\lim\limits_k\int_{[0,a]}f(kx) = ar$ where $f:[0,\infty) \to \mathbb{R}$, bounded, Lebesgue measurable, and $\lim\limits_{x\to\infty} f(x) = r$. $$ \int_{[0,a]}f(kx) = \int \chi_{[0,a]}(x)f(kx) ...
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1answer
27 views

Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$ \int f d\mu $$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
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40 views

Deduce that $f=0 \operatorname{a.e.}$

Let $f:[a,b]\to \mathbb R$ be a measurable function .Then Prove that if $\int _c ^d f(x)\operatorname {dx}=0$ for all $a\le c <d\le b$ then $f=0 \operatorname{a.e.}$ My try: Let ...
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1answer
16 views

Summation of integral.

Let $(E,\tau,\mu)$ be a measure space and $f:E\to \mathbb{R}$ is an absolutely integrable function, that is $$\int_{E} |f| \ \mathrm{d}\mu <\infty.$$ Set $A_n=\left\{x\in E \mid ...
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0answers
92 views

Proving Fatou type lemma

Let $f_1, f_2, \cdots$ and $f$ be nonnegative lebesgue integrable functions on $\mathbb{R}$ such that $$\lim_{n \to \infty}\int_{-\infty}^y f_n(x)dx = \int_{-\infty}^y f(x)dx \; \; \text{ for each ...
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3answers
54 views

$|f| $ is Lebesgue integrable , does it implies $f$ is also? [duplicate]

If $ f $ is Lebesgue integrable then $|f|$ is Lebesgue integrable but does the converse of the result is also true?
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1answer
49 views

If Darboux (Riemann equivalent), then Lebesgue?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be compact, define $$D^+(f):= \inf\left\{\int t:t\geq f, t= \text{step function}\right\}$$ $$D^-(f):= \sup\left\{\int t:t\leq f, t= \text{step ...
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3answers
60 views

Lebesgue Dominated Convergence Theorem example

For $x>0$ we have defined $$\Gamma(x):= \int_0^\infty t^{x-1}e^{-t}dt$$ Im trying to use Lebesgue's Dominated Convergence theorem to show $$\Gamma'(x):=lim_{h\rightarrow ...
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1answer
26 views

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$? [closed]

If $|f_n| \to 0$ and $f_n$ are integrable, is it true that $\int |f_n| \to 0$?
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1answer
20 views

Show $\lim\limits_k \int_{A_k} f_k = \lim\limits_k\int_A f_k,\;$ given $f_k \in\mathcal{L}^1(\mathbb{R}^n),\; \lim\limits_k\lambda(A_k\Delta A) = 0$.

Show $\lim\limits_k \int_{A_k} f_k = \lim\limits_k\int_A f_k,\;$ given $f_k \in\mathcal{L}^1(\mathbb{R}^n),\; \lim\limits_k\lambda(A_k\Delta A) = 0$. Here $\{A_k\}$ and $A$ are Lebesgue measurable. ...
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0answers
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
24 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
23 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
22 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 $$ \frac{\sqrt2}{\sqrt \pi}\frac{x}{x^2+ s^2}$$ ...
0
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1answer
31 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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0answers
36 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
19 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
56 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
64 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 ...
0
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1answer
23 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
49 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
65 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
22 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
81 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
45 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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0answers
19 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
36 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 ...
3
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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
37 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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0answers
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
33 views

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

Let $\,f, f_n $ be Lebesgue integrable functions mapping reals to extended reals such that, almost everywhere, $\,f_n \to f $. Show that ...
1
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1answer
32 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 ...
0
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1answer
20 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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0answers
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
votes
1answer
33 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 ...
1
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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 ...