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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2
votes
1answer
38 views

Sequence of integrable function with $\sum_{n=1}^\infty \|f_n\|_1<\infty$. Show that $\sum_{n=1}^\infty f_n$ converges a.e. and is integrable.

Let $\{f_{n}\}$ be a sequence of functions in $L^1(\mathbb{R})$ such that $\displaystyle \sum_{n=1}^\infty\|f\|_{1}<\infty.$ Show that $$f(x): = \sum_{n=1}^\infty f_n(x)\text{ converges a.e., }\, f\...
0
votes
0answers
14 views

Generalizing integral identity from characteristic function to $f\in L^1$

I read the proof, on F.J. Jones, Lebesgue integration on Euclidean space, of the fact that, if $u:[a,b]\to\mathbb{R}$ is an absolutely continuous non-decreasing function (therefore differentiable ...
3
votes
1answer
29 views

A strong version of the Dominated Convergence Theorem

Let $(X, \Sigma, \mu)$ be a measure space, and let $f, f_n:X\rightarrow \mathbb{C}$ be measurable functions with $f_n\rightarrow f$ pointwise. Assume that there are integrable functions $G, g_n:X\...
0
votes
2answers
43 views

Estimate $\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x$ without spherical coordinates.

Is it possible to estimate the following Lebesgue integral ($\|\cdot\|$ is the 2-norm) $$\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x, \, x\in\Bbb R^d$$ in terms of $\delta$ when $\delta\to 0$? ...
0
votes
2answers
90 views

Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$ \mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?. $$ No ideas how to start this one. I see that the limit of ...
1
vote
2answers
42 views

$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
7
votes
2answers
89 views

Can the Substitution Rule be Interpreted as a “Change of Measure”?

I just started learning measure and rigorous integration theory on my own along side my calculus class and I've noticed that with the substitution rule, you have something that looks like this $$ \int^...
3
votes
3answers
29 views

Why does $A_1^\text{c}$ have an infinite number of measurable subsets?

Let $\mathcal{A}$ be a $\sigma$-algebra. Show that if $|\mathcal{A}| = \infty$, then $\mathcal{A}$ is uncountable. We want to construct an infinite sequence of nonempty disjoint measurable sets. ...
5
votes
2answers
37 views

Assuming $\sum_{n = 1}^\infty \int |f_n| < \infty$, properties that follow for integral

How do I see that if $\sum_{n = 1}^\infty \int |f_n| < \infty$, then $\sum_{n = 1}^\infty f(x)$ converges absolutely almost everywhere, is integrable, and its integral is equal to $\sum_{n = 1}^\...
4
votes
1answer
57 views

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying some properties, prove $f\equiv 0$ a.e.

Let $f$ be a Lebesgue measurable function on $\Bbb{R}$ satisfying: i) there is $p\in (1,\infty)$ such that $f\in L^p(I)$ for any bounded interval $I$. ii) there is some $\theta \in (0,1)$ ...
1
vote
0answers
53 views
+50

Following conditions for convergence of measures equivalent

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Let $\mu_n$ be a sequence of finite measures on $([0, 1], \mathcal{B})$ and let $\mu$ be another finite measure on $([0, 1], \mathcal{B})$. ...
3
votes
1answer
32 views

Does it follow that $\mu$ is a measure? [duplicate]

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$. ...
3
votes
1answer
32 views

Alternate proof of the dominated convergence theorem by applying Fatou's lemma to $2g - |f_n - f|$?

Here is a proof of the dominated convergence theorem. Theorem. Suppose that $f_n$ are measurable real-valued functions and $f_n(x) \to f(x)$ for each $x$. Suppose there exists a nonnegative ...
7
votes
1answer
37 views

Variant of dominated convergence theorem, does it follow that $\int f_n \to \int f$?

Suppose $f_n$, $g_n$, $f$ and $g$ are integrable, $f_n \to f$ almost everywhere, $g_n \to g$ almost everywhere, $|f_n| \le g_n$ for each $n$, and $\int g_n \to \int g$. Does it follow that $\int f_n \...
4
votes
1answer
27 views

$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty$ implies $\{f_n\}$ is uniformly integrable?

Suppose $\mu$ is a finite measure and for some $\gamma > 0$, we have$$\sup_n \int |f_n|^{1 + \gamma}d\mu < \infty.$$Does it follow that $\{f_n\}$ is uniformly integrable?
6
votes
1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
2
votes
1answer
21 views

seq. of nonneg. Lebesgue measurable functions on $\mathbb{R}$, have $\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx$?

Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$If not, ...
5
votes
2answers
61 views

$f: \mathbb{R} \to \mathbb{R}$ integrable, $F(x) = \int_a^x f(y)\,dy$, $F$ necessarily continuous

Suppose $f: \mathbb{R} \to \mathbb{R}$ is integrable, and we define$$F(x) = \int_a^x f(y)\,dy.$$Why does it follow that $F$ is necessarily a continuous function?
0
votes
1answer
29 views

Prove $(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$

Let $\text K(x)$ be a Cantor function on $[0,1]$ prove $$(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$$ here $(\text L)$ denotes Lebesgue-integral. Attempt: ...
8
votes
1answer
83 views
+50

Intuition behind proof of bounded convergence theorem in Stein-Shakarchi

Theorem 1.4 (Bounded convergence theorem) Suppose that $\{f_n\}$ is a sequence of measurable functions that are all bounded by $M$, are supported on a set $E$ of finite measure, and $f_n(x) \to f(x)$ ...
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
1
vote
1answer
39 views

Evaluate $\left(\mbox{L}\right)\int_{0}^{1}\,\mathrm{ g}\left(x\right)\,\mathrm{d}x$

Let $\,\mathrm{g}\left(x\right) = \left\lbrace\begin{array}{rl} \,\mathrm{f}\left(x\right)\,, & x \in \mathbb{Q} \\[1mm] -\,\mathrm{f}^{2}\left(x\right)\,, & x \not\in \mathbb{Q} \end{...
0
votes
1answer
54 views

Do we have $\|x\|_p \le \|x\|_q$, where $x$ is bounded and $p>q$? [closed]

Assume that $x\in L_p(0,\infty) \cap L_q(0,\infty)$, where $p>q$. Do we have $$\|x\|_p \le \|x\|_q?$$ where $x$ is bounded.
-1
votes
1answer
45 views

How to prove that an $L^2$ function is also an $L^1$ function? [closed]

I have a function $f(t)$ defined on $[-a,a]$ that belongs to $L^2$. How do I prove that $f$ also belongs to $L^1$? In general this fact is not true. Is the cauchy-schwarz inequality the only way? If a ...
1
vote
2answers
14 views

Dealing with a Sequence of Sets with Two Indices and Simple Function based on that Sequence of Sets

Okay so I have a measurable function $f$ and a set $E_{n,i}$ $$E_{n,i}=\left\{ x:\frac{i-1}{2^{n}} \leq f(x)<\frac{i}{2^{n}}\right\}$$ where $i=1,...,n2^{n}$ and $n=1,2,...$ Then I have another ...
2
votes
0answers
21 views

Show that $E=\cup_{k=1}^{\infty}E_k$, where for each index $k, E_k$ is measurable, and $(f_n)$ converges uniformly to $f$

Let $(f_n)$ be a sequence of measurable functions on $E$ that converges to the real-valued $f$ pointwise on $E$. Show that $E=\cup_{k=1}^{\infty}E_k$, where for each index $k, E_k$ is measurable, and $...
4
votes
1answer
66 views

Lebesgue integration by substitution

I read that, if $f\in L^1[c,d]$ is a Lebesgue summable function on $[a,b]$ and $g:[a,b]\to[c,d]$ is invertible and such that $g\in C^1[a,b]$ and $g^{-1}\in C^1[a,b]$, then $$\int_\limits{g([a,b])}f(x)\...
2
votes
1answer
83 views

Find an appropriate function such that the composition is Lebesgue integrable on $[0,1]$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a measureable function. Show there exists an $\omega:\mathbb{R} \rightarrow\mathbb{R}$ such that $$\lim_{t \rightarrow \infty}\omega(t)=\infty$$ and $g(t)=\...
0
votes
0answers
16 views

On a second set of calculations for the Buchstab function

In this post I've added simple calculations deduced for the Buchstab function as claims. Question. Can you choose the worst of these claims and show where were my mistakes or inaccurancies? Also ...
1
vote
1answer
34 views

A neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which $\lim_R\int_{-R}^R|f|dx<\infty$

Is there a neat characterization of measurable functions $f:\mathbb R\rightarrow \mathbb C$ for which the limit of Riemann integrals satisfies $\lim_R\int_{-R}^R|f|dx<\infty$ in terms of elements ...
2
votes
0answers
91 views

Area and coarea formula and its application, change of variables

I have a question about an application of area and coarea formula and change of variables. Let $D$ be a bounded and connected open subset of $\mathbb{R}^{d}$ with $C^{1}$-boundary. Define $D_{\...
0
votes
2answers
72 views

Prove that a limit of an integral function is finite

I have to prove that the following limit is finite \begin{equation} \lim_{x \to b}\int_a^x \left(\int_\xi ^x (b-y)^{-\alpha}e^{-2y} dy \right)(b-\xi)^{-(1-\alpha)}e^{2 \xi} d \xi \end{equation} I'm in ...
2
votes
1answer
73 views

$\int_X f^p d\mu = p\int_{[0,+\infty)} t^{p-1}\mu(\{x\in X: f(x)>t\}) d\mu_t$ for any natural $p\ge 1$

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra ...
6
votes
0answers
88 views

Spectacular failure of Lebesgue differentiation for rectangles

Let $\mathcal{R}$ be the set of rectangles in the plane and, given $f \in L^1$ let $$ f^*(x) = \sup_{x \in R \in \mathcal{R}} \frac{1}{ \lvert R \rvert} \int_R \lvert \, f \,\rvert $$ as defined in ...
0
votes
1answer
17 views

Show: If $u \in W^{1,p}(B_4(0))$ then $\int_{B_4(0)} \frac{|\nabla u|^p}{|x-a|^{n-1}} dx < \infty$ for a.e. $a \in B_1(0)$.

I'm currently working through the article "Topology and Sobolev Spaces" by Brezis and Li and as basis for the proof of an important result the following fact is used: If $u \in W^{1,p}(B_4(0))$ ...
2
votes
1answer
25 views

Continuity of a characteristic function of a translated set

Let $E \subseteq \mathbb{R}$ be a measurable set. Is it true that $\chi_{E+t}(x) \rightarrow \chi_{E}(x)$ as $t \rightarrow 0$, where $E+t = \{x+t \, | \, x \in E\}$ for each $t \in \mathbb{R}$, ...
0
votes
0answers
14 views

$\int_V\frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2}d\mu_{\boldsymbol{x}}$ on infinite cylinder containing $\boldsymbol{r}$

Let $V\subset \mathbb{R}^3$ be a solid infinite cylinder, or cylindrical shell of radii $R_1<R_2$, and let $\boldsymbol{r}\in\overline{V}$ be any point belonging to the interior, or the surface, of ...
0
votes
3answers
95 views

An equality concerning the Lebesgue integral

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra ...
3
votes
1answer
50 views

Problem 5, Chapter 3 from Stein and Shakarchi's “Real Analysis” on a version of the FTC

The problem reads: Suppose that $F$ is continuous on $[a,b]$, $F'(x)$ exists for every $x\in(a,b)$, and $F'(x)$ is integrable. Then $F$ is absolutely continuous and $$F(b)-F(a)=\int_a^b F'(x)\,...
0
votes
1answer
15 views

Sequence of Functions on $[0,1]$ with Derivatives Bounded by $L^1$ Function

I'm stuck on the last step of a real analysis/advanced calculus problem and could really use some help. The problem is as follows: Let $f_n$ be continuously differentiable on $[0,1]$ satisfying, for ...
2
votes
1answer
81 views

About $ \int_{(0,1)^n} \frac{1}{x_1^{\alpha_1}+x_2^{\alpha_2}+…+x_n^{\alpha_n}} dm_n $

I'd like to prove the following two results, but besides the "trivial" implications, I haven't been able to crack them: $$ \int_{(0,1)^n} \frac{1}{x_1^{\alpha_1}+x_2^{\alpha_2}+...+x_n^{\alpha_n}} ...
2
votes
1answer
54 views

Is this integral continuous? (with respect to $z$)

Consider the integral $$\int_0^\infty f(t)e^{tz}\,dt,$$ where $f$ is an integrable function. Is this integral continuous with respect to $z$ (complex variable) on the domain $\{z=x+yi:x<0,y\in\...
1
vote
0answers
69 views

How to prove $f_n \in L^1$

I was trying to build a scheme to solve this kind of question: Let $D$ be a domain of $\Bbb R^n$ and $f_n\colon D \to \Bbb R$. Say if $f_n \in L^1(D)$. First of all I need to check that both $f_n$ ...
1
vote
1answer
61 views

Show that integral is analytic

Let $h:[0,\infty)$ be an integrable function. Prove that the function $$g(z)=\int_0^\infty h(t)e^{tz}\,dt$$ is analytic on $\{z=x+yi:x<0,y\in\mathbb{R}\}$. How do I start for this question? I ...
0
votes
0answers
26 views

Integral of magnetic field inside cylinder

Let $V\subset\mathbb{R}^3$ be an infinitely high solid cylinder, or a cylindrical shell of radii $R_1<R_2$, whose axis has the direction of the unit vector $\mathbf{k}$. For any point of ...
1
vote
0answers
22 views

Justify $\lim_{n\to\infty}n^p\int_0^1\sum_{k=n}^\infty\frac{\sigma(k)e^{x/k}}{k^{p+2}\log\log k} dx=\frac{e^\gamma\int_0^1f(x)dx}{p}$

Inspired in PROBLEM 207, La Gaceta de la Real Sociedad Matemática Española, Vol. 16, N0. 3 (page 507 in spanish, proposed and solved by Furdui), I've tried write examples of this new statement ...
0
votes
2answers
32 views

Lebesgue integral question (double integral)

Let $g,h$ be nonnegative Lebesgue measurable functions on $\mathbb{R}$. Prove that $$\int_{-\infty}^\infty g(x)^2h(x)\,dx=\int_0^\infty\int_{\{t\in\mathbb{R}:g(t)>x\}}2h(t)x\,dtdx.$$ I am lost on ...
1
vote
1answer
24 views

Dilation convergence in L^1

Below is a question, which I asked before, from Stein's Real Analysis. I've provided a partial solution, which I think it's pretty along the lines of what needs to be done, however, I have no ...
3
votes
1answer
49 views

If $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|\max (0, \log |f_n|)<\infty$, then $f_n\to f$ in $L_1$

I'm going through old analysis qualifying exams, and have come to a roadblock on the following problem: Suppose that $\{f_n\}\subset L_1([0,1])$, $f_n\to f$ pointwise, and $\sup_{n} \int_{0}^{1} |f_n|...
6
votes
1answer
151 views

How to start an eigenvalue problem

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems... Please give me a hint or a clue, cause I ...