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
20 views
Conditions on a measure
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>0$ $t^s\in L^1(\mathrm d\mu(t))$, but functions $\mathbf{1}_{t>0} $ and $\mathbf{1}_{t\in (0,1)}$ are not necessarily in ...
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2answers
47 views
Convergence of $\int_{A_n} f$ to $0$
I am looking for a name or a reference in a textbook for the following result in order to quote it.
For any $f\in L^1(\mathbb{R})$-integrable function, we have
$$\lim_{n\to\infty}\int_{A_n} ...
2
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1answer
52 views
Please check my proof (that integral = 0 implies integrand = 0 a.e. in a Bochner space setting)
Let $V$ be Hilbert and separable. Suppose $f \in L^2(0,T;V')$.
I want to show that if
$$\langle f, v \rangle=0$$
holds for all $v \in L^2(0,T;V)$, then
$$\langle f(t), w(t) \rangle_{V',V} = ...
1
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1answer
27 views
proving that a certain function is absolutely continuous
Let's consider positive real numbers $\alpha,\beta>0$. Then let's define the function:
$ f(x)= x^{\alpha} sin(\frac{1}{x^\beta})$ if $x\in (0,1]$.
$f(0)=0$.
Prove that if $\alpha>\beta>0$ ...
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2answers
31 views
Contradiction to Continuity of Integration?
For each of the two functions $f$ on $[1,\infty)$ defined below, show that $\lim_{n \rightarrow \infty} \int_1^n f$ exists while $f$ is not integrable over $[1,\infty)$. Does this contradict the ...
1
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1answer
38 views
Showing measurability of a function (integrals and Bochner space)
Suppose $V$ is Hilbert space and I know that for every $u \in L^2(0,T;V)$, the integral
$$\int_0^T \langle f(t),u(t)\rangle_{V^*,V}$$
exists (because it equals another integral, and I know that ...
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1answer
19 views
Does N-L formula holds for everywhere differentiable function?
Suppose that $f$ is differentiable on $[a,b]$ everywhere, if $f'(x)$ is Lebesgue integrable on $[a,b]$, can we say that the Newton-Leibniz formula holds for $f$?, More precisely, dose the following ...
3
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1answer
57 views
Integral inequality using positive and negative parts
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a measurable (with respect to the Lebesgue measure), $\pi$-periodic function which is Lebesgue integrable over $[0,\pi]$.
Moreover assume that ...
-1
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0answers
54 views
Measure of continuous set-valued mappings
Let $m: \mathcal{B}(\mathbb{R}^n) \rightarrow [0,1]$ be a probability measure without point masses.
Consider a continuous set-valued, measurable-valued, mapping $S: \mathbb{R}^m \rightrightarrows ...
1
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1answer
53 views
Lebesgue vs. Riemann integrable function
While trying to learn the difference between Lebesgue and Riemann integrals, I came across the following example:
$$\int_{0}^{1}t^\lambda\,\mathrm dt$$
What I know so far:
only for $\lambda>0$ ...
8
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1answer
97 views
Limit of measurable functions in finite measure space
Let $(X,\mathcal{M},\mu)$ be a measure space with $\mu(X)<\infty$. Let $f_n$ be a sequence of measurable real-valued functions such that $f_n$ converges pointwise a.e. to a real-valued function ...
2
votes
1answer
37 views
Continuity of a probability integral
Let $m: \mathcal{B}(\mathbb{R}^n) \rightarrow [0,1] $ be a probability measure without point masses.
Let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be (jointly) continuous.
Define ...
2
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0answers
55 views
An almost orthogonality principle for $L^p$
If two functions are far from being orthogonal, their difference cannot be too large in $L^2$. A precise statement (easily verified with the Pythagorean theorem) is as follows: let ...
2
votes
2answers
76 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} ...
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1answer
41 views
Continuity of a probability integral with supremum
Let $m: \mathcal{B}(\mathbb{R}^n) \rightarrow [0,1] $ be a probability measure without point masses and let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be continuous. Let $\epsilon ...
4
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1answer
69 views
Explicitly calculating Lebesgue Integral
I'm studying Lebesgue theory and when a problem asks to actually perform a computation I'm at a loss at what to do. Is the typical way to proceed to observe that an integral is the same as its Riemann ...
1
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1answer
38 views
Generalization of Fatou's Lemma
Fatou's Lemma: Let $\{f_n\} \rightarrow f$ pointwise a.e. on $E$, then $\int_E f \leq \liminf \int_E f_n$.
Generalization: Prove that if $\{ f_n \}$ is a sequence of nonnegative measurable functions ...
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0answers
57 views
Is there a solution manual for Royden fourth edition?
I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices.
I have the solution manual for the third edition. Is there ...
0
votes
2answers
47 views
function in $L^1\setminus L^2$
I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded).
Does anyone know such a function?
2
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1answer
34 views
Lebesgue–Stieltjes integral from 0 to $\infty$ on $\mathbb{R}^+$
In the Stochastic analysis course we encountered the following integral
$\int_0^\infty H^2_sd[M,M]_s$,
where $H_s$ is a predictable process, $M_s$ is a uniformly integrable martingale in $L^2$, ...
0
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0answers
53 views
evaluate integral
$\def\sgn{\mathop{\mathrm{sgn}}}$Let $$f(t) =- \int_0^t\sgn f(s)\,ds \;,t \geq 0 \tag{$*$} $$
with $$\sgn(x)=\begin{cases}1 & x > 0\\-1 & x \leq 0\end{cases} $$
Assume that ...
0
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1answer
48 views
Question about absolute continuous functions (preservation of null sets)
I'm trying to prove that a function $ f:[a,b] -> \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 ...
0
votes
2answers
39 views
Integrating over a triangle
Let $\hat T$ be the triangle spanned by $(0,0)$, $(1,0)$ and $(0,1)$. Let
$$I(r,s) = \int_{\hat T} x^r y^s d(x,y)$$
with $r,s\in\mathbb N\cup\{0\}$.
Prove that
$$I(r,s) = \frac{r!s!}{(2+r+s)!}$$
To ...
0
votes
1answer
41 views
Non-negativity in Fatou's Lemma and Lebesgue Dominated Convergence Theorem
Im currently busy with Measure Theory and noticed that the main theorems that I want to use, require the non-negativity condition.
Fatou's Lemma has the condition that we need a sequence $\{f_n\}$ ...
0
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0answers
29 views
Integration by part for Lebesgue integral
I tried to prove one theorem in convergence of random variables and found myself in a little bit of trouble when doing integration by part. The reason being it involves a Lebesgue integral which I am ...
2
votes
1answer
43 views
Lebesgue integration and countable partitions
The book "A Primer of Lebesgue Integration" by H.S. Bear defines lebesgue integration through lower and upper sums $L(f,P) = \sum m_i\mu(E_i)$ and $U(f,P)=\sum M_i\mu(E_i)$ where infinite countable ...
1
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1answer
79 views
$\liminf, \limsup$, Measure Theory, show: $\lim \int n \ln(1+(f/n)^{1/2})\mathrm{d}\mu=\infty$
Let $(X,\Omega,\mu)$ be a measure space and $M^+(X,\Omega)$ denote the set of all non-negative real valued measurable functions.
If $f \in M^+(X,\Omega)$ and $0< \int f \mathrm{d}\mu < ...
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2answers
61 views
prove $ F(x)=\int_0^\infty {\sin(tx)\over(t+1)\sqrt t} dt \in C^\infty(\mathbb R^*) $
prove that :
$$ F(x)=\int_0^\infty {\sin(tx)\over(t+1)\sqrt t} \, dt \in C^\infty(\mathbb R^*) $$
i end up proving that $F(x)\in C^ \infty(\mathbb R^{*+})$ not $\mathbb R^*$ , and i studied the ...
2
votes
1answer
58 views
Tails of family of integrable functions
It is well known that tail of an integrable function on $\mathbb{R}^d$ is small, i.e., Given $\epsilon>0$, there is $R>0$ such that $$\int_{\{|x|>R\}}|f(x)|dx<\epsilon.$$
I was wondering ...
1
vote
1answer
54 views
Measurability is true?
Can you please help me solve this on measurablilty? My TA did not go over this in measurability. He said we are not going over this but you can do this if you want. Can someone please explain to me. ...
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0answers
39 views
Measurablilty with infinity
Can someone please show me how to solve this wth measurability? My TA did not show us this yet but I was curious how to solve it.
I know that $1<x<n$ for ${\frac{1}{x^2}}$ and the $sup$ $=$ lim ...
0
votes
1answer
38 views
Embedded Lp spaces [duplicate]
Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that ...
2
votes
1answer
53 views
Calculate the Riemann Stieltjes integral
This is not a homework question. It is a past exam question and I would appreciate some step by step help, as I never understood this concept in class.
Let $\alpha(t) = n^2$ for $t\in[n,n+1).$ ...
1
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2answers
29 views
Showing that an absolute integrable monotone decreasing function $f: [1,\infty[ \rightarrow \mathbb{R}$ is in $L^p([1,\infty[)$
For an exercise in my analysis course, I have to show that: if $f: [1,\infty[ \rightarrow \mathbb{R}$ is monotone decreasing and $f \in L^1([1,\infty[)$, then $f \in L^p([1,\infty[)$ for every $p > ...
0
votes
2answers
46 views
Riemann integral and Lebesgue integral
$f:R\rightarrow [0,\infty)$ is a Lebesgue-integrable function. Show that
$$
\int_R f \ d m=\int_0^\infty m(\{f\geq t\})\ dt
$$
where $m$ is Lebesgue measure.
I know the question may be a little dump.
...
1
vote
1answer
67 views
Proving that for $f\geq0$ on $X$, $\int_X f d\mu = 0$ iff $f = 0$ a.e.
Okay, so the question is the following:
Suppose $f \geq 0$ is a measurable function on the measure space $(X,\Sigma,\mu)$. Prove that
\begin{align} \int_X f d\mu = 0 \text{ if and only if } f = 0 ...
1
vote
0answers
62 views
For what $p$ is $x^p$ Lebesgue Integrable?
Revising for an exam on Monday any help with the following question would be greatly appreciated;
If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...
1
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1answer
27 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
0
votes
1answer
38 views
Show that E is measurable?
Suppose $E_1= [1, 1 \frac12] , E_2 = (2, 2\frac14), E_3 = [3, 3\frac18], E_4 = (4 , 4 \frac{1}{16}) , \dots , E= \bigcup_{n=1}^{\infty}E_n
$
i) Show $E$ is measurable
ii) Compute $m(E)$
Here is ...
0
votes
1answer
181 views
Function from $L^{2}(0,T; L^{2}(\Omega))$
Suppose I have a function $u\in L^{2}([0,T]\times\Omega)$ for some bounded domain $\Omega$ in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. I managed to prove that this implies $u\in L^{2}(0,T; ...
1
vote
1answer
27 views
A limit of integrals
Let $f:[0,T]\to\mathbb{R}$ be a Lebesgue integrable function. For each $h>0$ we define the piecewise function $f_h$ by
$$f_h(t)=f(h\left[\frac{t}{h}\right])\quad\mbox{for}\quad t\in[0,T].$$
Can we ...
0
votes
1answer
57 views
E measurable with m(E) < $\infty$?
Suppose that $E$ is measurable with $m(E)$ $<$ $\infty$.
ii) Show that $\displaystyle \ \ \int_E 2f\,\,\,$ $=$ $2$$\displaystyle \ \ \int_E f\,\,\,$ if $f$ is bounded and measurable.
I told my ...
0
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1answer
81 views
If $f :\mathbb{R}\to\mathbb{R}$ is measurable, then $E = \{x: f(x) \geq 3\}$ is measurable
Prove: Suppose $f : \mathbb{R}\to\mathbb{R}$ where $f$ is measurable and $E = \{x: f(x) \geq 3\}$. Show $E$ is measurable.
I saw this statement while reading in a paper and thought this might ...
2
votes
1answer
41 views
About the induced vector measure of a Pettis integrable function
In what follows, $X$ stands for a Hausdorff LCTVS and $X'$ its topological dual.
Let $(T,\mathcal{M},\mu)$ be a finite measure space, i.e., $T$ is a nonempty set, $\mathcal{M}$ a $\sigma$-algebra of ...
2
votes
1answer
37 views
Evaluating a Gaussian Integral
How to prove that
$$\int_{\mathbb{R}^N}e^{-\langle Ax,x\rangle}\operatorname{dm}(x)=\left(\frac{\pi^N}{\det A}\right)^{\frac{1}{2}}$$
Where $A:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is a symmetric ...
2
votes
0answers
28 views
Lebesgue Integral Rudin Problem [duplicate]
Suppose {$n_k$} is an increasing sequence of positive integers and E is the set of all x$\in$($-\pi, \pi$) at which {sin$n_k x$} converges. Prove that $m(E)=0$.
Hint: For every A $\subset$ E, ...
2
votes
0answers
32 views
Question about Lebesgue integration on $\mathbb{R}^N$
Let $\Omega\subseteq\mathbb{R}^N$ be an open set and $f:\Omega\to[0,+\infty[$ a measurable function, bounded over each compact $K\subset\Omega$. If there is a $C>0$ such that
...
0
votes
2answers
44 views
Proving that $L^p \subset L^q$ when $1 \le q \le p$ [duplicate]
Let $(E,\mathcal{F},\mu)$ be a measure space such that $\mu(E)=1$ and let $L^p=L^p(E, \mathcal{F},\mu)$. Prove that
$L^p \subset L^q\text{ if } 1 \le q \le p$.
I let $f \in L^p$. Then $(\int_E ...
3
votes
0answers
53 views
How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space
Let $\mathcal{L}^2[(0,1)]$
denote the set of $\mathbb{C}$-valued square integrable functions on the interval [0,1].
Prove that $\mathcal{L}^2[(0,1)]$ forms a Hilbert Space.
I believe that I can ...
0
votes
0answers
47 views
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
let $f(x,y)=\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}$.
Show that
$\int_{0}^{1} dx \int_{0}^{1} f(x,y) dy=\frac{\pi}{4}$
$\int_{0}^{1} dy \int_{0}^{1} f(x,y) dx=-\frac{\pi}{4}$
