0
votes
1answer
28 views

Existence of Monotone Sequence of Simple Functions

Let $\Omega$ be a measurable space with measurable sets $\Sigma$ and denote the space of simple functions by:$$\mathcal{S}:=\{s:\Omega\to\mathbb{R}:s=\sum_{k=1}^K ...
0
votes
1answer
24 views

Measurability of Modulus

Context: This problem came up while reading an essay on Bochner integrability. Let $\Omega$ be a measure space and $E$ a Banach space. Consider two plain functions $f:\Omega\to E$ and $g:\Omega\to ...
1
vote
0answers
17 views

Integration by parts for multidimensional Lebesgue-Stieltjes Integrals

I am concerned with the following problem: I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given ...
1
vote
1answer
49 views

Property of the Riemann Integral

Hello fellow Mathematics enthusiasts. I was hoping someone could help me with the following problem from Terry Tao's Introduction to Measure Theory: Let $[a,b]$ be an interval, and let $f,g:[a,b] \to ...
0
votes
2answers
49 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
0
votes
2answers
46 views

Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
5
votes
1answer
86 views

What is an example of a function that is measurable but not strongly measurable?

Let $(\Omega, \Sigma)$ be a measurable space and $X$ a Banach space. Let $f: \Omega \rightarrow X$. $f$ is called measurable if every the preimage of every Borel set in $X$ is an element of ...
0
votes
1answer
28 views

About a $\sigma$-finite measure

Consider a probability space $(\Omega,\mathcal H,P)$ and a real random variable $X$ such that $E(X)$ is well defined (also infinite values are allowed). Is it true that the measure ...
1
vote
1answer
32 views

From $\left\lVert \sup_{M>M_0} \left| \sum_{k=M_0}^M f_k \right| \right\lVert_2 < \epsilon$ show convergence a.e. of the series.

I'm having trouble with the following 'qual' problem. For one, I don't know what to make of the absolute value inside the $L^2$-norm. In short, I just don't have any intuition for it. And I don't ...
1
vote
1answer
32 views

Continuity of measure and integration

Suppose that f is a measurable function $(\Omega, \mathfrak{F}, \mu)$ such that $\int_{A}f \, d\mu \geq 0 \forall A \in \mathfrak{F}$. Prove that $f \geq 0 \ \mu$-almost surely. Hint: Let $A_n = ...
1
vote
1answer
45 views

Halmos Measure Theory section 39 Theorem D

I have trouble explaining the remark "The function $\phi$ plays the role of Jacobian (or, rather, the absolute value of the Jacobian) in the theory of transformation of multiple integrals". I know ...
0
votes
0answers
31 views

Derivation using Ito calculus?

I am reading the paper "Coupling Wiener processes by using copulas" by P. Jaworski and I've come across a statement I cannot reproduce. Let $L^{-}$ and $L^{+}$ be differential operators acting on ...
2
votes
0answers
35 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
1
vote
1answer
18 views

Reference on Riesz representation theorem for $L^p(0,T,X)$ spaces.

Brezis Functional Analysis book proves the following Riesz representation theorems for usual $L^p(\Omega)$ spaces: In what book can we find an analogous of these theorems for $L^p(0,T,X)$ spaces? ...
6
votes
1answer
136 views

Separability of a set with norm $\thickapprox$ $L^1$ +$L^{\infty}$

Let $(M, \mathcal{A}, \mu)$ a complete separable probability space. Recall that complete means that any subset of a measurable set with zero measure is measure (and has zero measure) and separable ...
2
votes
2answers
56 views

Is a $L^1$-function which is linear near the origin in $L^p$?

Suppose you have a function $f$ on $\mathbb{R}$, such that $$\int_{-\infty}^{\infty} | f(x) | \, \mathrm{d} x < \infty$$ and $$\int_{-u}^u |f(x)| \, \mathrm{d} x = \mathcal{O}(u)$$ for $u \to 0$. ...
1
vote
1answer
44 views

Series converges but term by term integration not valid?

Give an example of a series $\sum g_n$ of Lebesgue integrable functions on $\mathbb{R}$ that converges but for which term by term integration is not valid. This is last minute exam revision so I do ...
-1
votes
1answer
23 views

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$?

What is the integral of $\int_{\mathbb{N}} s d\mu$ where $\mu$ is the counting measure on $\mathbb{P}(\mathbb{N})$? What does it mean for $s$ to be integrable? 1. This is last minute exam revision. ...
2
votes
1answer
56 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
1
vote
1answer
60 views

How to understand uniform integrability?

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?
0
votes
1answer
42 views

An issue with $\infty \cdot 0$ in showing that Cartesian product of a set with a null set has measure zero

Here is the problem: Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be $\sigma$-finite measure spaces. Furthermore $A\in \mathcal A$ and $N\in \mathcal B$ such that $\nu(N)=0$. Let ...
0
votes
1answer
51 views

If two measures are equal, are the integrals with respect to these measures equal?

If $\mu$ and $\nu$ are probability measures such that $\mu=\nu$, then is it true that for all measurable function $f$ $$\int fd\mu=\int fd\nu \ \ \ ?$$ It is true for integrable functions but if $ ...
17
votes
1answer
432 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
0
votes
1answer
68 views

Negative part of the integrand in an iterated integral

Hi everyone: Suppose that $(X,\mathfrak{M},\mu)$ and $(Y,\mathfrak{N},\nu)$ are two measure spaces and $f(x,y)$ is an extended real valued measurable function on $X\times Y$. Suppose we can not apply ...
1
vote
1answer
44 views

The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers. We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under ...
1
vote
0answers
17 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
0
votes
1answer
37 views

A tricky integral with vanishing domain

I would love to have the following result, however I got no clue if it is even true! Let $B_n:=\{y:\varepsilon_n<|y|\leq\tilde{\varepsilon}_n\}$ for some sequences ...
0
votes
0answers
17 views

Finiteness of the lower integral implies finiteness a.e. of the function

I want to prove that if a function $f$ is $\mu$-measurable, $f\geq 0 $ $\mu$-a.e., then the integral of $f$ exists, that is its upper and lower integrals coincide. I've found the proof in Modern and ...
0
votes
1answer
44 views

If $f$ is $+\infty$ on a set of positive measure and the integral exists in $[-\infty,+\infty]$, must the integral be $+\infty$?

Suppose $(X,\mathcal{M},\mu)$ is a measure space and $f$ a measurable function from $X$ to $[-\infty,+\infty]$. Suppose that $$\int_{X}f\ d\mu$$ exists in $[-\infty,+\infty]$, and that $X$ contains a ...
3
votes
0answers
64 views

Why is the value assigned to a gauge integral well defined (unique)?

Why is the value assigned to a gauge integral well defined (unique)? If we would have given a net (so an underlying order that happens to be directed), then the limit would be unique given a ...
1
vote
1answer
36 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
3
votes
2answers
69 views

Without Lebesgue

Everyone knows following problem. Let $f$ be positive function on $[0,1]$ and there exist $I = \int_{0}^{1}f(x)dx$. Prove that $I>0$. (recall that there are only two cases: $I=0$ or $I>0$. NOT ...
1
vote
1answer
32 views

variation of a function over countable intervals

Let $f$ be a function of bounded variation on $[0,1]$. Let $\{[a_n,b_n]\}_{n=1}^\infty$ such that $(a_n,b_n)$ are pairwise disjoint and $\cup_{n=1}^\infty [a_n,b_n]=[0,1]$. (for example, $[1/2, 1], ...
1
vote
1answer
33 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
3
votes
0answers
119 views

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
6
votes
0answers
170 views

Open problems in Federer's Geometric Measure Theory

I wanted to know if the problems mentionned in this book are solved. More specifically, at some places, the author says that he doesn't know the answer, for example :"I do not know whether this ...
2
votes
1answer
51 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
1
vote
2answers
47 views

Jordan measure zero discontinuities a necessary condition for integrability

The following theorem is well known: Theorem: A function $f: [a,b] \to \mathbb R$ is Riemann integrable if and only if its set of discontinuities has Lebesgue measure zero. Now if we change ...
2
votes
2answers
93 views

Do we need the $f,g \geq 0$ condition for $\int f \ d\mu = \int g \ d\mu$?

My lecture notes state the following corollary: Let $f,g \in \mathcal M_\bar{{\mathbb R}}$ (that is, numerical measurable functions), $f=g$ $\mu$-almost everywhere and $f,g \geq 0$. Then $\int f \ ...
4
votes
1answer
39 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
1
vote
1answer
50 views

Haar measure on $G \times G$, where $G$ is compact

Let $G$ be a compact group. Let $\mu'$ and $\mu$ be the Haar measure on $G \times G$ and $G$, respectively, and further such that $\mu'(G \times G) = 1$ and $\mu(G)=1$. Does it follow that $\mu' = \mu ...
0
votes
0answers
34 views

Basic facts related to Haar measure

I have a compact group $G$ and continuous functions $f_1, f_2$ from $G$ to $\mathbb{C}$ and $g: \mathbb{R} \rightarrow \mathbb{C}$. I have two questions related to Haar meausure. Is it true that $$ ...
4
votes
2answers
48 views

p-norm of a function

Let $f\in L^1(\mu)\cap L^\infty(\mu)$. I have proved for any $1<p<\infty$, $f\in L^p(\mu)$, $w(p)=||f||_p$ is continuous w.r.t. $p$, and $\lim_{p\to \infty}||f||_p=||f||_\infty$. Is $w(p)$ ...
3
votes
1answer
63 views

Integration using Lebesgue dominated convergence theorem

This is an old comp question I'm working on. $$\lim_{n\to\infty}\int_{[0,1]}\frac{d\lambda}{x^\frac{1}{n}(1+\frac{x}{n})^n}$$ I am having trouble finding a dominating function. Thinking about the ...
2
votes
1answer
37 views

Product measure and integrals of simple functions

Let $(\Omega_1 , \mathcal{X}, \mu)$ and $(\Omega_2 , \mathcal{Y}, \nu)$ be two $\sigma$-finite measure spaces, and let $\mu \times \nu$ be product measure on the $\sigma$-algebra $\mathcal{X} \times ...
1
vote
3answers
54 views

Positive integral everywhere implies positive function a.e

I would like to get feedback on my demonstration of this simple statement : Let $f$ be an integrable function on the measure space $(X,S,\mu)$. \begin{align} \text{If }\int_E f \, d\mu \geq 0\text{ ...
4
votes
1answer
68 views

Functions with every point being a Lebesgue point

For a locally integrable function $f$ a point $x$ is a Lebesgue point if the integral averages of deviations from $f(x)$ over balls centered at $x$ converge to $0$ as the balls shrink to the point. ...
-1
votes
4answers
101 views

Are null sets necessarily closed?

Hi everyone: Is a null set of $\mathbb{R}^n$, $(n>0)$, necessarily closed? Give a counter example. Thanks for your reply.
2
votes
1answer
69 views

An amazing inequality of the integration of two functions.

Let $f:[0,1]\longrightarrow\mathbb{R}$ be measurable and $g\in L^1[0,1]$ such that for all $t>0$, $$ \int_{|f(x)|>t}|g(x)|~\mathrm{d}x\leq \frac{3}{t^2}. $$ Prove that for $1<p<2$, $$ ...
0
votes
1answer
26 views

Integrability of the logarithm wrt a finite Borel measure

I have a finite Borel measure $d\phi$ on $(0,1)$, i.e. $\int_0^1 d\phi(x) < \infty$. Is it also true that $\int_0^1 \log (x) d\phi(x) < \infty$? The function $\log$ is integrable at 0, so ...