0
votes
1answer
8 views

Calculating a limit of integrals

I am having a problem with the following exercise: Show that for every bounded borelian function $\varphi : \mathbb{R} \rightarrow \mathbb{R}$, $\underset{n}{lim} \frac{n}{\sqrt{2\pi}} ...
2
votes
1answer
27 views

On convergence a.e and convergence measure

I have a question. First, I know that convergence in measure of a sequence of functions $f_n$ is different than convergence a.e., wich means there are sequences that converge in measure but not a.e. ...
0
votes
1answer
38 views

Name for LDC: Lebesgue?

Is there also a name associated to the Lebesgue dominated convergence theorem like Beppo-Levi or Fatou? Would Lebesgue be reasonable? Who did originally prove it?
1
vote
0answers
39 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
2
votes
2answers
72 views

What is the motivation to build measure theory?

I started reading about measure theory on wikipedia, and downloaded some PDFs, but they all start defining things that I can understand, but can't imagine the motivation to define these things. ...
1
vote
0answers
19 views

Product of counting measure and the integral

Given the sigma algebra $P(\mathbb{N}^2)$(or $P(\mathbb{Z}^2)$ and counting measure $n$, I need to show that $n \times n$ is the counting measure for the aforementioned sigma algebra and compute the ...
-1
votes
0answers
55 views

Complex Measures: Lebesgue Decomposition

Disclaimer: This thread is related to: Singular Continuous Measures: Definition? Context Let $\Omega$ be a measure space with finite measure $\mu<\infty$. Consider a finite measure ...
1
vote
1answer
21 views

Does integrating out a variable in a two-variable measurable function produce a measurable function?

This problem is not a mere consequence of Fubini’s Theorem, so I thought that it would be suitable for posting here on MSE. Let $ (X,\Sigma,\mu) $ and $ (Y,\text{T},\nu) $ denote $ \sigma $-finite ...
-2
votes
2answers
53 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
0
votes
1answer
38 views

Lebesgue Mean Value Theorem

Disclaimer: This proof is taken out from Rudin, Real and Complex Analysis. Let $\Omega$ be a finite measure space $\lambda(\Omega)<\infty$. Denote the mean value by: ...
0
votes
1answer
26 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
1
vote
1answer
27 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
0
votes
1answer
18 views

Example about Dominated Convergence Theorem

So I was reading my textbook about Dominated Convergence Theorem: I have $(X,\mathscr{F},\mu)$ as a measure space I have $f,f_n,: X\to [-\infty, \infty], g:X\to [0,\infty]$ integrable and it is the ...
2
votes
2answers
36 views

A measure is sigma-finite if, and only if, there exists a integrable function w such that its image is contained in (0,1)

I have to prove the following proposition: Consider a measure space $(S,\Sigma,\mu)$. Prove that $\mu$ is $\sigma$-finite if, and only if, there exists $w\in\mathcal{L}^1(S,\Sigma,\mu)$ such that ...
3
votes
0answers
41 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 ...
0
votes
0answers
44 views

$f$ absolutely convergent on $\mathbb{N}$ iff integrable

I am looking for some hints on the following Let $X$ be the set of all positive integers, $\Sigma$ the class of all subsets of $X$, and $\mu(E)$ (for any $E\in \Sigma)$ the number of points in $E$. ...
2
votes
1answer
47 views

continuous-time version of fatou's lemma

I have just read a textbook on stochastic processes that implicitly uses the fact that \begin{equation} \int \liminf_{t \to \infty} f_t \leq \liminf_{t \to \infty} \int f_t, \end{equation} for ...
1
vote
1answer
37 views

How is Fubini Theorem used here?

Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in ...
5
votes
2answers
96 views

Which integral is larger?

The question: Given $f$ to be a positive, measurable function on $[0,1]$, which is larger, $\displaystyle\int_0^1 f(x)\log f(x)\,dx$ or $\displaystyle\left(\int_0^1f(s)\,ds\right)\left(\int_0^1\log ...
0
votes
2answers
83 views

Complex Measures: Total Variation Measure Decomposition

Let $\mu:\Sigma\to\mathbb{R}$ be a complex measure. Does the total variation measure admit a decomposition: $$\mu(E)=|\mu|(E\cap A)-|\mu|(E\cap B)+i|\mu|(E\cap C)-i|\mu|(E\cap D)$$ with $A,B,C,D$ ...
0
votes
1answer
25 views

diffeomorphisms preserve zero measure

Suppose $\Omega\subset \mathbb R^N$ is an open set and $f:\Omega\rightarrow f(\Omega)$ is a $C^1$ diffeomorphism. Show that if $F \subset \Omega$ has zero measure then $f(F)$ has zero measure. I ...
1
vote
0answers
59 views

Complex Measures: Variations

I highly doubt a proof of mine for complex measures... Construction A complex measure can be decomposed into positive measures: ...
1
vote
1answer
48 views

Complex Measures: Integration

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! Integration w.r.t. complex measure usually is defined via the Radon-Nikodym ...
0
votes
1answer
51 views

Spectral Measures: Integration

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
1
vote
1answer
31 views

Lesbegue integration w.r.t scaled measure

I was wondering if we have a measure $\mu$ and $a \in \mathbb{R}$ with $\lambda = a\mu$ if we get : \begin{align} \int fd\lambda = a\int fd\mu \end{align} It seems to work for the basic definition of ...
6
votes
2answers
71 views

A pseudometric on the space of the measurable functions is complete

I'm working in the following exercise: Suppose $(X, \mathcal A, \mu)$ is a finite measure space and suppose $\mathcal F$ is the set of all $\mathcal A$-measurable functions $f: X \rightarrow \mathbb ...
2
votes
2answers
54 views

Spectral Measures: Support vs. Concentration

The support of a Borel spectral measure is defined by: $$\lambda\in\mathrm{supp}E:\iff E(U)>0\quad\lambda\in U\in\mathcal{T}$$ (See the german wikipedia article: Spektralmaß) Now, consider a Borel ...
1
vote
1answer
62 views

Spectral Measures: Integration of Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
1
vote
1answer
66 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
1
vote
2answers
73 views

Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
1
vote
1answer
33 views

Lebesgue Dominated Convergence: Alternative Proof?

Is there an alternative proof of Lebesgue's dominated convergence theorem relying on positive functions only? The point is I'd like to prove that for positive functions: $$\int ...
0
votes
0answers
32 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
1
vote
1answer
32 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
0
votes
1answer
34 views

Vitali Set: Inner Measure vs. Outer Measure

Context Nonlinearity in general of the Lebesgue integral for nonmeasurable functions reduces in some sense to inner and outer measure of nonmeasurable sets: ...
0
votes
1answer
43 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
27 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
34 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
53 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
52 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
50 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
95 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
30 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
34 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
34 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
34 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? ...
7
votes
1answer
143 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
59 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 $u \to 0$. ...