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
12 views

Proving that $\left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3$, given $1\leq a<b$

If $1\leq a<b$, then $$ \left|\int_{a}^{b}\frac{\sin(x)}{x}dx\right|\leq 3.$$ Proceeding by integration by parts; let $u(x)=\sin(x)$ and $dv(x)=1/x$, then $u'=\cos(x)$ & $v(x)=\log(x)$. We ...
1
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0answers
50 views

Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero

Let $A$ be a Lebesgue measurable subset of $\Bbb R$. 1) Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. 2) ...
0
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2answers
51 views

Explicit Integrals and LimInf/LimSup

(a) Show that $f(t):=\int_0^\infty e^{-tx}\frac{sin \space x}{x}dx$ exists for $t>0$ and defines a differentiable function $f$. Calculate $f'(t)$ for $t>0$ and evaluate it explicitly. (b) Prove ...
2
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2answers
34 views

Part of proof to show Lebesgue-lebesgue measurable

I want to prove the following: Suppose $E$ is a subset of $\Bbb R$, let $\gamma(E)=\{ (x,y)\in \Bbb R \times \Bbb R :x-y\in E\}$. If $E\in \Bbb B$ (Borel/Lebesgue measurable set), show that ...
0
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1answer
31 views

Lebesgue integrals and polinomial functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a non zero polinomial function. Prove that $f\notin\mathbb{L}(\mathbb{R})$.In other words is not Lebesgue integrable.
2
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0answers
14 views

Approximate an integrable function using a simple function (Proving existance)

Let $f \in L^1(\mathbb{R})$, and let $\epsilon > 0$. Show that exists simple function $g=\sum_{k=1}^{n}c_k 1_{A_k}$, such that, $$\int_\mathbb{R} |f(x)-g(x)|dx \leq \epsilon$$,and such that $n \in ...
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2answers
30 views

Helping understand line integral $\int_{K,+}{(x+y)}dx+(y-x)dy$

I have a huge problem with understanding line integrals and would be much obliged for your help! We have: $$\int_{K,+}{(x+y)}dx+(y-x)dy$$ and the following parameterization: ...
0
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2answers
20 views

Inequality of Lebesgue integrals

Let $f,g\in\mathbb{L}(E)$. Suppose that $f\leq g$ and $A:=${$x\in E| f(x)<g(x)$}. Prove that $\int_{E}f<\int_{E}g$ if and only if $A$ has positive measure.
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0answers
9 views

If $| \int_A fdλ| ≤ λ(A)$ for $A \subset [-1,1]$. then range of f contained in [-1,1]

Let f : [−1, 1] → R be a continuous function. Let λ be the Lebesgue measure on [−1, 1]. Suppose $| \int_A fdλ| ≤ λ(A)$ for all measurable sets A ⊆ [−1, 1]. I want to show that the range of f is ...
-2
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1answer
19 views

Integrable functions in $\mathbb{R}$? [on hold]

Let $f\in\mathbb{L}(\mathbb{R})$ integrable. If $a>0$, prove that $f^{-1}((a,+\infty))$ has finite measure.
2
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1answer
22 views

Continuous functions are locally integrable?

If $K\subset\mathbb{R}$ is compact and $f:K\rightarrow\mathbb{R}$ continuous then $f\in\mathbb{L}(K)$. In other words $f$ is integrable in $K$. So far i know that since $f$ is continuous then $f(K)$ ...
2
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1answer
26 views

Application of Dominated Convergence Theorem help finding a Dominating function

$$\lim_{n\to\infty}\int_0^\infty \frac{n\sin(x/n)}{x(1+x^2)}$$ I wish to use the Lebesgue Dominated Convergence theorem to solve this, but I'm having trouble finding a dominating function, $g(x)$. ...
0
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1answer
24 views

small shift in domain of simple function

Suppose $s:[0,1]\to\mathbb R$ is a simple function with $s(0)=s(1)$. Let $S:\mathbb R \to \mathbb R$ be $S(x)=s(x-\lfloor x\rfloor)$, the function that repeats $s$ on each interval $[n,n+1]$. I'm ...
-3
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0answers
29 views

If $f = g +h$ then $\int_E f = \int_E g + \int_E h$ is independent of the choices of $g$ and $h$ [on hold]

Let $f$ be a measurable function on $E$ which can be expressed as $f = g +h$ where $g$ is a finite and integrable function over $E$ and $h$ is nonnegative over $E.$ Define $\int_{E} f = \int_E g + ...
3
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1answer
54 views

Part of proof of the set of continuous integrable functions is dense in $L^1(\Bbb R)$

I want to prove: If $g$ belongs to $L(\Bbb R, \Bbb B, \lambda)$ and $\epsilon\gt 0$, then there exists a continuous function $f$ such that $\Vert g-f\Vert_1=\int \lvert g-f\rvert \,\text{d}\lambda \lt ...
0
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1answer
17 views

Holders Inequality: Suppose $\int_{0}^\infty x^{-2}|f|^5 dx < \infty$. Prove that $\lim_{t \to 0} t^{-\frac{6}{5}} \int_0^t f(x)dx = 0$

I discovered last night that I have an error in my proof to the following problem and I need help fixing it (or need a new solution) $$ \text{Suppose that} \int_{0}^\infty x^{-2}|f|^5 dx < \infty. ...
0
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1answer
13 views

Inequality regarding measure of function and integral of function

Let $(X,\Sigma,\mu)$ be a measure space. Let $f$ be a measurable function and $t > 0, t\in \mathbb{R}.$. Denote: $$C_f(t) = \mu \{x \in \Omega : |f(x)| \geq t \}.$$ In the first part of ...
1
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1answer
26 views

Justifying the differentiation property of the Fourier transform

Let the Fourier transform of $f\in L^1(\Bbb R)$, denoted by $\mathcal{F}f$, be defined as $$ \mathcal{F}f(y) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ixy} f(x)\,dx.$$ An oft-quoted result ...
2
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2answers
41 views

If a measurable function $f$ has zero integral over every measurable set *of finite measure*, then $f=0$ a.e.?

Let $X$ be a locally compact Hausdorff space, and let $\mu$ be a regular measure on $X$. Suppose that $g : X \to \Bbb C$ belongs to $L^{\infty}(X)$. My question is : Is it sufficient to assume ...
2
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4answers
155 views

Show $\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$

It's claimed that $$\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$$ by first expanding $\frac{\log(1-x)}{x}$ into a power series and then doing term-by-term integration. I want to justify this by ...
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0answers
34 views

Approximating integrable functions

I wish to prove the following If $f$ is integrable and $f:\mathbb{R}\to\mathbb{R}$ then $$\lim_{t\to0}\int_{-\infty}^{\infty}|f(x)-f(x+t)| = 0$$ I have in my notes that if there exists a $g(x) \in ...
3
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0answers
63 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
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0answers
10 views

Convergence of two Lebesgue-Stieltjes integrals

I have I have a collection of bounded variation and right-continuous functions, $(F_n)_{n \in \mathbb{N}}$, and another bounded variation and right-continuous function, $G$, which satisfy $$\sup_x ...
1
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1answer
66 views

Part of proof of term-by-term integration

I want to prove the theorem of term-by-term integration for lebesgue integrable functions (denoted as $L^1$ functions): Suppose $(g_n)$ is a sequence of $L^1$ functions over a measure space $(X,\sigma ...
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0answers
13 views

Is it a change of variable?

Hi everyone: In a book I am reading, they make a sort of "substitution" like this: let $B(0,R)$ be a ball in $\mathbb{R}^{N}$ $(N\geq2)$ and $f$ a locally integrable function. Let $\mu$ be a finite ...
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0answers
15 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 ...
1
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2answers
41 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), ...
10
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1answer
610 views

Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
3
votes
1answer
36 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 ...
2
votes
1answer
79 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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0answers
30 views

If $f'$ of a continuous $f$ exists except on a countable set $E$ and $f'$ is Lebesgue integrable over $[a, x]$ then $f(x)-f(a)=\int_a^x f'dx $. [duplicate]

If $f'$ (derivative of continuous function $f$) exists except on a countable set $E$ and $f'$ is Lebesgue integrable over $[a, x]$ then $f(x)-f(a)=\int_a^x f'dx $. Of course I'm aware of the proof ...
0
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0answers
18 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 $g$ ...
2
votes
1answer
27 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 ...
0
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0answers
53 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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0answers
15 views

Prove $T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$ where $T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$ is linear. [closed]

Prove $T\int_c^d f(x,y)dy = \int_{c}^dTf(x,y)dy$ where $T:\mathcal{C}[a,b] \to \mathcal{C}[a,b]$ (set of continuous functions on $[a,b]$) is linear and continuous in $\mathcal{L}^1$ norm, and ...
4
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2answers
43 views

Is $f(x)\exp(-x^2)$ summable if $f$ is square summable?

Suppose that $f \in L^2(\mathbb R)$; i.e. $$ \int_{- \infty}^\infty \vert f(x) \vert^2 dx < \infty. $$ Can we from this infer that $$ \int_{- \infty}^\infty \vert f(x)\vert e^{-x^2} \, dx < ...
1
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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) ...
0
votes
1answer
19 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 ...
1
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2answers
762 views

Question about absolute continuous functions (preservation of null sets)

I'm trying to prove that a function $ f:[a,b] \to \mathbb{R} $ is absolutely continuous iff $ \mu(A) = 0 \implies \mu( f(A)) = 0$ for all such $A \subseteq [a,b]$. I'm quite stuck. I'm trying to work ...
1
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0answers
59 views

Fourier transform of distribution

Let $f\in S_{\infty}$ be a Schwartz function and let us define a linear functional,for any $\varphi\in S_{\infty}$, $S_{\infty}\to\mathbb{C}$, $\varphi\mapsto (f,\varphi)$ ...
3
votes
1answer
38 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 ...
0
votes
1answer
14 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 ...
1
vote
3answers
54 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 ...
4
votes
3answers
48 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?
2
votes
1answer
41 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 ...
2
votes
2answers
130 views

Lebesgue integral over a collection of sets

Let $E$ and $\langle E_n \rangle$ be measurable sets in $\mathbb{R}$. Suppose that $f$ is Lebesgue integrable over $E$. If $E_n\subset E$ for all $n$ and $\displaystyle \lim_{n\to \infty} ...
1
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1answer
26 views
1
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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 ...
0
votes
1answer
19 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. ...