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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9
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0answers
208 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded Lipschitz domain and $u$ is a measurable function. A sufficient condition for the integral $\int\limits_{\Omega}{uvdx}<\infty,\forall v\in ...
7
votes
0answers
379 views

Egorov's theorem for this Lebesgue integral

I want to prove Egorov's theorem using this Lebesgue integral defined by the upper integral $$\int^*f:=\left\{\int h ; h \ge f \text{ and h upper-continuous }\right\}$$ $$\int_*f:=\left\{\int h ; h ...
6
votes
0answers
81 views

Various integration theories

Could anyone briefly explain, or point me towards a resource explaining, the main differences between the main integration theories, namely: Riemann Integration Riemann-Stieltjes Integration ...
5
votes
0answers
53 views

Show that if $f$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$.

Let $A$ be a bounded measurable subset of $\mathbb{R}$. Show that if $f:A\rightarrow \mathbb{R}$ is measurable then $\{x\in A:c=f(x)\}$ is measurable for each $c$. Choose real $c$. Since $f$ is ...
5
votes
0answers
57 views

$L^2$ convergence of this sequence

I am given the following sequence of functions $(f_m)_{m \in \mathbb{N}}$. They are defined by $$ f_m(x):=\left( \frac{e^{-ix}-1}{-ix} \right)^m \left( \sum_{l \in \mathbb{Z}} \frac{\left|e^{-ix}-1 ...
5
votes
0answers
118 views

Convergence of the solution of Volterra integral equation with convergent kernel.

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where $g(t)$ and $K_n(t,s)$ are known(continuous) and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, ...
5
votes
0answers
138 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
5
votes
0answers
591 views

Lebesgue Integration fundamental questions

My question involves the definition of the Lebesgue integral. Most colloquial definitions I've read follow (2), in that f*(t) is the "length" of one of the horizontal rectangles and dt is the ...
4
votes
0answers
47 views

Prove that $f\in L^2$ and $\lim_{n\rightarrow\infty} \int_A f_n = \int_Af$

Let $A$ be a bounded, measurable susbset of $\mathbb{R}$. Prove that if $(f_n) \subset L^2 (A)$ converges uniformly to $f$ on $A$, then $f\in L^2(A)$ and $\lim_{n\rightarrow\infty} \int_A f_n = ...
4
votes
0answers
24 views

If $P$ and $Q$ are Lebesgue partitions then is $P\cap Q$ a refinement to $P$ and $Q$?

My teacher said that if $P$ and $Q$ are Lebesgue partitions then $P\cap Q$ is a refinement to $P$ and $Q$. But shouldn't $$P\cup Q$$ be the refinement to both $P$ and $Q$? Or, are both $P\cup Q$ and ...
4
votes
0answers
81 views

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
4
votes
0answers
61 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
4
votes
0answers
64 views

Represent total variation of continuous function by integration of counting function

$f : [a,b] \to \mathbb R$ is continuous, let $M(y)$ be the number of points $x$ in $[a,b]$ such that $f(x)=y$. prove that $M$ is Borel masurable and $\int M(y)dy$ equals the total variation of $f$ on ...
4
votes
0answers
70 views

Fubini–Tonelli theorem

By Fubini-Tonelli theorem, if one of $\int(\int |f(x,y)| dy)dx$, $\int(\int |f(x,y)| dx)dy$, $\int\int |f(x,y)| dxdy$ is finite , then $\int(\int f(x,y) dy)dx = \int(\int f(x,y) dx)dy=\int \int ...
4
votes
0answers
139 views

An a.e.-defined derivative which is not Lebesgue integrable on any interval?

If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost ...
4
votes
0answers
76 views

How is Riemann–Stieltjes Integration insufficient for developing modern probability theory?

If we consider Riemann–Stieltjes integration then it can perfectly account for mixed probability distribution (a continuous R.V with some point mass). So why would we still need Lebesgue Integration ...
4
votes
0answers
196 views

Why do people apply Fubini-Tonelli theorem so easily?

I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage. This text seemingly covers various topics on measure theory, but i think that's it. This text ...
4
votes
0answers
216 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
3
votes
0answers
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 ...
3
votes
0answers
12 views

$q$-integral of product of two functions

How to solve $\int\limits_a^b \left[\int\limits_t^b (x-tq)_{\Re(\alpha n+\beta-1)} d_qx\right]|\phi(t)|d_qt$ and what will be the answer? Where $|q|<1$ and the function $\phi$ be in the space ...
3
votes
0answers
83 views

Expectation of a random vector and Fubini's theorem

I have a question related to the definition of expectation of a random vector, in particular, to its relation (if any) with Fubini's Theorem. Consider the random vector $X:=(X_1,X_2,X_3)$ of ...
3
votes
0answers
23 views

Interchanging Malliavin derivative with Lebesgue integral

I am reading Oksendal's book "Malliavin calculus for Levy processes with application to finance". In the proof of Lemma 4.9 (page 47), the author interchanges the Malliavin derivative $D_t$ with the ...
3
votes
0answers
51 views

When are $\frac{1}{|x|^s}$ and $\log|x|$ integrable near the origin?

When are $\frac{1}{|x|^s}$ for $s>0$ and $\log|x|$ integrable near the origin? I'm reading Evans PDE and in the construction of the fundamental solution of Poisson's equation, he defines $$ \Phi(x) ...
3
votes
0answers
67 views

Calculate $\lim\limits_{n \to \infty} \int_0^1 \frac{n^p x^r\log x}{1 + n^2 x^2}dx$

The problem is to find an integrable function that bounds $f_n(x) = \frac{n^p x^r \log x}{1 + n^2 x^2}$ where $r>0$, $p<\min \{2,1+r\}$ so we can calculate $$\lim_{n \to \infty} \int_0^1 ...
3
votes
0answers
153 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 $1\leq p< \infty$ ? ...
3
votes
0answers
60 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 ...
3
votes
0answers
117 views

Upper Lebesgue sum with a new partition

Assume we have a $f$ from $R$ to $[0, \infty)$, which is Lebesgue integrable. Show that there exists a sequence of bi-infinite partitions $Y_n$ of the $y$-axis for which the Lebesgue upper sum is ...
3
votes
0answers
41 views

Riemannian Volume vs. Euclidean Volume

If $M$ is a Riemannian manifold and $\phi$ is a Borel measurable function into $M$, then what is the relationship between $\int_M \phi \,d\mu$ and $\int_{\mathbb{R}^d} \psi\circ \phi\, d\lambda$; ...
3
votes
0answers
492 views

Proving translational invariance of Lebesgue integral

I am asked to show that the Lebesgue integral is invariant under translations. Specifically, Let $(\mathbb{R}, \Sigma, \mu)$ be a measure space, and for any $f:\mathbb{R}\rightarrow\mathbb{R}$ ...
3
votes
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 ...
3
votes
0answers
56 views

Reversing an “inverse Fourier transform”

Let $g$ be the Fourier transform of an unknown function $y\in L_1(-\infty,\infty)$:$$g(\lambda)=\int_{\mathbb{R}}y(x)e^{-i\lambda x}d\mu_x$$Let $f$ be defined as ...
3
votes
0answers
48 views

Measurability of $f:X\times Y\to\mathbb{K}$ and $f(-,y):X\to\mathbb{K}$

Let $(X,\mu_x)$ and $(Y,\mu_y)$ be two measure spaces endowed with $\sigma$-additive compete measures $\mu_x$ and $\mu_y$, respectively. Let $\mu:=\mu_x\otimes\mu_y$ be the Lebesgue extension of ...
3
votes
0answers
143 views

Equivalence of Lebesgue integral definitions

I'm currently enrolled in a course in integration and functional analysis following Avner Friedman's Foundations of Modern Analysis. However, I noticed that his definition of the Lebesgue integral is ...
3
votes
0answers
71 views

Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$.

Let $I = [a, b], E \subset I, m(E) = 0$ (but $E$ not empty). Prove that there exists a continuous increasing function $\sigma(x)$ on $I$ such that $\sigma'(x) = + \infty$ for every $x_0 \in E$. I am ...
3
votes
0answers
90 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
votes
0answers
38 views

Is this a decomposition of the same function?

Let's say we have some integral, such that for a particular function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ $$\int_{\mathbb{R}^{n-m}} \int_{\mathbb{R}^m}f^+ - ...
3
votes
0answers
100 views

$|f(t) - f(s) |\leq \int_s^t g $ then $f(t) - f(s) = \int_s^t h.$

Let $f : [0,1] \rightarrow [0, + \infty)$. If there exists $g \in L^1([0,1]) $ s.t. for every $t,s \in [0,1]$ holds $$ |f(t) - f(s)| \leq \int_s^t g(u) \, du \quad (t>s),$$ then there ...
3
votes
0answers
106 views

Show: $\int f\, d\mu=\sum\limits_{y\in f(X)}y\cdot\mu(f^{-1}(y)))$

Let $f\colon X\to\overline{\mathbb{R}}_{\geq 0}$ be a measurable function which only takes a countable number of values. Show that $$ \int f\, d\mu=\sum\limits_{y\in ...
3
votes
0answers
83 views

Pseudo norm-exercice

Let $f$ be a measurable function with finite values almost everywhere. We put $$N_0(f) = \displaystyle\int \dfrac{|f|}{1 + |f|} d \mu.$$ We denoted by $L^0$ the set of measurable functions $f$ such ...
3
votes
0answers
284 views

Prove Heisenberg uncertainty principle (measure and integration theory)

Here is a question in measure and integration theory, Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
3
votes
0answers
412 views

Riemann integral vs Lebesgue integral

Let $f$ be analytic on a domain $\Omega$ of the complex plane, such that the closed disc $\overline{D(0,R)}$ is contained in $\Omega$. What is the difference between $$ \int_{D(0,R)}|f(w)|dm(w)$$ and ...
2
votes
0answers
26 views

Why is the Newton quotient measurable when the conditions are like the following.

Let $f(x, y), 0 \le x, y, \le 1$, satisfy the following conditions: for each $x$, $f(x, y)$ is an integrable function of $y$. $\displaystyle\frac{\partial{}f(x, y)}{\partial{}x}$ is a bounded ...
2
votes
0answers
50 views

Understanding principal value integral

I'm reading the original article on distance covariance (link), and throughout the article the author uses the following lemma: Can someone please explain what he actually means by "principal value ...
2
votes
0answers
84 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) ...
2
votes
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 ...
2
votes
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 = ...
2
votes
0answers
32 views

Dominated convergence theorem vs continuity

Let $\{f_n\}$ be a sequence of functions in $L^2(0,1)$ such that $\lim_n f_n = f$ pointwise and $\vert f_n(x) \vert \leq g(x)$ for some integrable function $g$. By the dominated convergence theorem it ...
2
votes
0answers
19 views

Moment problem for order statistics

Problem: Let $X_{1},\cdots,X_{n}$ be samples from $X$. If $E\left|X\right|^{a}<\infty$ for some $a>0,$ and $n,k,r$ satisfies $r\leq a\cdot\min\left(k,n-k+1\right),$ then ...
2
votes
0answers
36 views

Differentiability under the integral sign of absolutely continuous $f(-,x)$

I read that if $f:[a,b]\times X\mapsto\mathbb{R}$, where $X$ is a space endowed with measure$\mu_x$ and we define in $[a,b]\subset\mathbb{R}$ the usual Lebesgue linear measure $\mu_t$, ...
2
votes
0answers
77 views

The integral $\int f \mu(dt)$ versus $\int f \mu(t) dt$

In the theory of continuous-time branching processes there is a quantity called the Malthusian parameter $\alpha \in \mathbb{R}$ defined by: $$ \int_0^{\infty} e^{-\alpha t} \mu(dt) = 1 $$ The ...