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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0answers
18 views

Why does Fubini's theorem not hold/apply to this function?

Given the function $$ f(x,y) = \begin{cases} e^{y-x}, x > y \geq 0 \\ -e^{x-y}, 0 \leq x \leq y \end{cases}$$ I have already determined that $$ \int_0^\infty \left( \int_0^\infty f(x,y) dx \right) ...
-2
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0answers
6 views

f(x) Lebesgue measurable and integrable, show g(x,y) = f(x)/x is wrt product measure [on hold]

f(x) Lebesgue measurable and integrable, show g(x,y) measurable and integrable is wrt Lebesgue product measure on (0,1 x (0,1) where g(x,y) = f(x)/x for 0< y< x<1 and 0 elsewhere
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0answers
10 views

Some questions on convergence of measurable functions and increasing sequences.

My question relates to the proof of existence of the essential supremum, on Planetmath (http://planetmath.org/?op=getobj&id=11400&from=objects). I have some qualms with the proof so would ...
0
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2answers
16 views

If $\lim_{n\to\infty} \int_E f_n dx$ is finite, prove $f$ is Lebesgue integrable on $E$ and $\lim_{n\to\infty} \int_E f_n dx=\int_E fdx$.

Let $\{f_n\}_{n=1}^\infty$ be a decreasing sequence of Lebesgue integrable functions defined on a measurable set $E$. Suppose there is a function $f$ such that $f_n(x)\to f(x)$ almost everywhere on ...
1
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1answer
37 views

Does the integral converge I can't find counterexample

I found the following question in the book of kolomogorov fomin introductory real analysis and I don't know how to solve it. Does anyone have any ideas? Suppose $f$ is integrable on sets ...
3
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1answer
43 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
1
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2answers
67 views

Do Riemann and Lebesgue integrals always agree?

I know that on a closed bounded interval, say $[a,b]$ in $R^1$, if a function is Riemann integral, then it is Lebesgue integrable, and the values of those two integrals are the same. But, is this ...
0
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1answer
25 views

A Quick Question on the Monotone Property of Integrals.

Let $(\Omega,\mathcal{A},\mu)$ be a measure space with $f$, $g\in \mathcal{L}_1(\Omega,\mathcal{A},\mu)$. If for any $A\in\mathcal{A}$ we have $$\int_Afd\mu\geq\int_Agd\mu\space ,$$ show that $f\geq ...
1
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1answer
35 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
0
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1answer
27 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
5
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3answers
55 views

Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$.

(Jones, p. 246) Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. This seems pretty easy to prove in the following way: Let $g_j$ be a ...
1
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1answer
58 views

I need to prove whether two sequences are equidistributed or not

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. A sequence $\{x_{n}\}$ in $[0,1]$ is called ...
0
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1answer
14 views

Finite measure space & sigma-finite measure space

A measure space $(X, \Sigma, \mu)$ is finite if $\mu(X)<\infty$. It is equivalent to saying that $(X, \Sigma, \mu)$ is finite if $\mu(E)<\infty$ for all $E \in \Sigma$ A measure space $(X, ...
1
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1answer
39 views

Finite meaure space with $f \in L^p$ [duplicate]

Given a finite measure space $(X,\Sigma,\mu)$, for $1<p<\infty$, if $f \in L^p(X)$, then $f \in L^1(X)$. Can anyone show me how to start the proof? Thanks.
1
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1answer
24 views

Any function in $L^p$ space is a linear combination of simple functions? True OR not?

Any function in $L^p$ space is a linear combination of simple functions for $1<p<\infty$. Is this true? So any function in $L^p$ is measurable. So any measurable function can be represented ...
0
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0answers
17 views

Reconstructing a measure from its (absolutely continuous) marginals

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation ...
2
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1answer
13 views

Question on the difference between a limit of an integral and if a function is integrable

In this thread I asked a question about getting started on a problem. The question is this: Let $f$ be a function such that $f(x) = \frac {(-1)^n}n$ for $x\in [n, n+1)$. 1) Show that $lim_{n\to ...
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1answer
44 views

Is f Lesbegue Integrable?

I've got a problem that I have been working on in my Real Analysis class, and am not sure on the answer. The problem is below along with my thoughts so far. Problem: Let f be a function such that ...
0
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1answer
18 views

Integrating with respect to a linear combination of two signed measures

Let $(X, d)$ be a metric space and $\mathcal{B}(X)$ the Borel $\sigma$-algebra of X. Let $\mu, \nu$ be two real-valued signed measures defined on $(X, \mathcal{B}(X))$ and $f : X \to \mathbb{R}$ Borel ...
0
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1answer
34 views

Congruent Sub-Intervals with Reimann-Integrable Functions

Let $f:[a,b]\to\Bbb R$ be a Riemann-integrable function. Prove that for each $\sigma\gt0$ there exists a partition $\mathcal P$ of $[a,b]$ into congruent sub-intervals(that is, $x_{j}=a+{j(b-a)\over ...
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0answers
28 views

a question about real analysis,I need to know whether these two sequences are equidistributed. [closed]

For each subset $S\subset [0,1]$ write $X_{s}$ for the "indicator function" as $X_{s}(t)=1$ when $t\in S$ and $X_{s}(t)=0 $ when $t\notin S$. a sequence $\{x_{n}\}$ in [0,1] is called ...
2
votes
3answers
89 views

Inherent Pitfall of Lebesgue Integration?

I am studying Real Analysis with Royden's Book. I noticed that for a function f differentiable almost everywhere on [a, b] and f' integrable over [a, b], it does not imply that $ f(x) = \int_{[a, ...
2
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1answer
48 views

a question about the evaluation of integral [duplicate]

Let $\alpha:[0,1] \to R$ be the Cantor function. Evaluate $$\int_{0}^{1}xd\alpha $$and $$\int_{0}^{1}x^2d\alpha.$$ I know that the Cantor function is continuous and monotone increasing, how can I ...
1
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0answers
51 views

Puzzles in a proof

From a previous link in MSE: Prove the set of which sin(nx) converges has Lebesgue measure zero (from Baby Rudin Chapter 11), the question states Suppose that $\{n_k\}$ is an increasing sequence ...
1
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1answer
47 views

Existance of limit and Integrability of a function

The questions is: Let $f$ be a function defined as $f(x) = (-1)^n/n $ for $x \in [n, n+1), n \in \mathbb{N}.$ Show that $lim_{n\to\infty}$ $\int_{[1,n]}\ f $ exists Also, is $f$ integrable on ...
0
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0answers
26 views

Show that Thomae Function is Lebesgue Integrable

I have the Thomas Function defined as follows: $f(x): [0,1] \to \mathbb R$ $f(x) = q$ if $x$ is rational and $x = p/q$, $0$ otherwise (please note that this is the usual definition of THomae's ...
0
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0answers
18 views

Integrability of a function, $(x,y)\mapsto 1_{[0,\infty)\times[0,\infty)}(x,y)(e^{-x}-e^{-y})$.

Is the function $$(x,y)\mapsto 1_{[0,\infty)\times[0,\infty)}(x,y)(e^{-x}-e^{-y})$$ integrable wrt. the lebesgue measure on $(\mathbb{R}^{2},\mathbb{B}_{2})$? I have shown that it's not integrable, ...
6
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3answers
82 views

Question about Dominated Convergence Theorem.

How to compute $$\lim_{n \to \infty}\int_0^{\infty}\Big(1+\frac{x}{n}\Big)^{-n}\sin\Big(\frac{x}{n}\Big) dx$$ I want to use the Dominated Convergence Theorem. so it becomes ...
2
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2answers
64 views

integral over a subset of $\mathbb {R}^2$ is not defined while…

consider the function $f(x,y)=\frac{xy}{(x^2+y^2)^2}$, we can see by some easy calculation that $\int_{-1}^1\int_{-1}^1 f(x,y)\,dx\, dy$ and $\int_{-1}^1\int_{-1}^1 f(x,y)\,dy\, dx$ exist and equals ...
2
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1answer
40 views

prove that $\int_{\Omega}|f_n-f_0|d\mu\rightarrow 0$ (By weaker assumption on Scheffé's lemma)

I'm dealing with this problem. Let $(\Omega,\mathcal{F},\mu)$ be a measure space and $\{f_n\}$ a sequence of nonnegative integrable functions. Suppose $f_n\xrightarrow{\mu} f_0$ and ...
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0answers
23 views

Are $L1$ functions with a.e. finite support a.e. equal to a continuous function?

I was wondering about this: Let $f \in L^1(\Omega)$ and $\Omega\subset \mathbb{R}^n$ be compact, then $f$ is the $L^1$ limit of continuous functions with support in $\Omega$. Egorov's theorem tells us ...
0
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1answer
40 views

prove the equivalence between a null set and a limit

I'm asked to prove that for any non-negative, measurable and integrable function $f$ on $[0,1]$, we have $\lim\limits_{a\to 0}\int_{0}^{a}fdx=0$. I want to use the theorem that for null set E, such a ...
2
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1answer
49 views

Let $g$ be a non-negative measurable function. Show $\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$.

Let $g$ be a non-negative measurable function. For $1 \leq p < \infty$, show that $$\int g(x)^p d\mu = \int_0^\infty p t^{p-1} m_g(t) dt$$ where $\mu$ is the Lebesgue measure and we are ...
2
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2answers
40 views

Comparing limits of integrals

If $$f_n:X\rightarrow [0,\infty]$$ is a sequence of measurable functions and we know that $$\lim_{n\rightarrow \infty }\int_X f_n \,d\mu=0,\qquad \qquad \tag{$\star$}$$ then can we conclude that ...
3
votes
1answer
75 views

Limit of the integral $\int_0^1\frac{n\cos x}{1+x^2n^{3/2}}\,dx$

Prove that $\displaystyle\int_0^1\frac{n\cos x}{1+x^2n^{\frac32}}dx\rightarrow0$ as $n\rightarrow\infty$. $f_n(x)=\frac{n\cos x}{1+x^2n\sqrt{n}}$ tends to zero function pointwise. It just ...
6
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1answer
40 views

how to prove $\int{f}d\mu=\sum_{x\in\Omega}f(x)$

Prove that $\int{f}d\mu=\sum_{x\in\Omega}f(x)$ when $f$ is absolutely summable, where $\mu$ is a counting measure on the measure space $(\Omega,\mathscr{F})$. Can someone give me hints?
0
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1answer
29 views

Definition of Lebesgue integrable function

If a function $f : \mathbb{R}^d \to [-\infty,\infty]$ is Lebesgue integrable, then by definition we have $$\int_{\mathbb{R}^d} |f(x)| \, dx < +\infty.$$ Is it possible to say that there exists a ...
4
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2answers
53 views

Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$

Prove $X$ is a complete vector space $\iff$ $\Big[\quad\sum_{n=1}^\infty \| x_n \| \implies \sum_{n=1}^\infty x_n$ converges$\quad\Big]$. I have seen a proof of this already in a lecture but I ...
6
votes
2answers
95 views

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e.

If $\int f=0$ and $f(x) \ge 0$ for all $x \in \mathbb{R}^d$, then $f=0$ a.e. I let $E \subset \mathbb{R}^d$ be a finite measurable set. I try to break this into two cases: Case 1: If $f(x)=0$ ...
1
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1answer
24 views

Dominated convergence and fundamental lemma of the calculus of variation

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
5
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1answer
88 views

Trying to calculate the integral limit $\lim_{n\rightarrow\infty} \int_{-\sqrt n}^{\sqrt n}\left (1 - \frac{x^2}{2n}\right)^ndx$

How to calculate following integral: $$\lim_{n\rightarrow\infty}\int_{-\sqrt{n}}^{\sqrt{n}}{\left(1-\frac{x^2}{2n}\right)^n}dx$$ Prove that this integral exists and compute its value. I just ...
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0answers
18 views

improvement of upper Lebesgue sum

In Pugh's real mathematical analysis, lower and upper Lebesgue sum are given as: $\underline{L}(f,Y)= \sum_\limits{i=1}^{\infty}y_{i-1}\cdot mX_{i-1}$ $\overline{L}(f,Y)= ...
1
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1answer
29 views

show Lebesgue dominated convergence theorem fails for ${n^2xe^{-nx}} x\in [0,1]$

show Lebesgue dominated convergence theorem fails for the sequence of functions $f_n=n^2xe^{-nx}$ $x\in [0,1]$ Here is my solution. Is it correct? $f_n$ is an integrable function the sequence ...
2
votes
2answers
35 views

Showing that $L^2\subset L^1$ for $L^2([0,t_f])$, with $t_f$ a fixed positive number.

I saw demonstrations using the Cauchy-Schwartz Inequality but I am still not convinced because the Inequality is as follows : $$ \left |\langle f,g\rangle\right | \leq \left \|f \right \|_{L_2} . ...
3
votes
0answers
120 views

Dense subset of $L^p[a,b]$

It is true that $C_{0}^{\infty}[a,b]$, (the space of all smooth functions f with the property that f and all its derivatives vanish at a and b) is dense in $L^p[a,b]$ with $1\leq p< \infty$ ? ...
2
votes
2answers
45 views

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?

If $f \in \mathcal{L}^{2}(\mathbb{R}^{n})$, does it imply that it is bounded almost everywhere?
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votes
2answers
27 views

Is this subspace of $L^1(\mathbb{R},m)$ closed? [closed]

Let $K$ be the subspace of $L^1(\mathbb{R},m)$ which contains precisely the functions such that $\int f=0$. Is $K$ closed? (EDIT: When I asked this question, I could only see that ${f:||f||_1=0}$ is ...
2
votes
0answers
46 views

Integration over a variety

If $ M $ is a differentiable manifold equipped with an Atlas $ \mathcal{A} = ( U_i , \varphi_i )_{ i \in I} $, we can then calculate the integral of a differential form $ \omega $ over $ M $ with the ...
1
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0answers
18 views

property of distribution function

Let $f$ a continuous map from $\mathbb{R} \rightarrow \mathbb{R}$ and let $L_1, L_2$ 2 probability measures on $\mathbb{R}$. Let $K$ be a closed set in $\mathbb{R}$. In a proof, I want to use the ...
1
vote
2answers
26 views

Application of dominated convergence theorem- find limit

Find (with justification) $$ \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx $$