1
vote
0answers
10 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 ...
-1
votes
0answers
21 views

Darboux integrable, $f$ continuous at x where g(x)=G(x) [on hold]

$f:[a_1,b_1]x[a_2,b_2]\rightarrow \mathbb{R}$ that is Riemann integrable, and let $g(x),G(x)$ functions with property $g(x)\leq f(x) \leq G(x)$, g=G a.e.! G(x), g(x) are obtain from proof Riemann int ...
0
votes
1answer
35 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
16 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
40 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
52 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
31 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
31 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
32 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
92 views
+50

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 ...
5
votes
0answers
136 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
45 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
41 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
91 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
33 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
48 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
32 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
44 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
48 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
34 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
48 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
59 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
99 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
67 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 ...
3
votes
1answer
116 views

When is $\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$

An elementary question on Riemann - Integration: Under what conditions on $f$ is the following true: $$\lim_{b\to a} \int_a^b f(x)dx=\int_a^af(x)dx=0$$ If $f$ is bounded in $[a,b]$, then this is ...
2
votes
1answer
24 views

Integrability in a domain of $\mathbb{R}^{N}$

Hi everyone: Let $f$ be a function defined on an open set of $\mathbb{R}^{N}$ $(N\geq1)$. Is there any difference between the following two statements? 1) $f$ is locally integrable 2) $f$ admits a ...
5
votes
0answers
51 views

Is $\sigma$-finiteness really a necessary condition for this problem?

Question: Let $(X, \mathcal A, \mu)$ be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given any $\varepsilon$, there exists a $\delta >0$ such that ...
3
votes
1answer
28 views

an argument that strengthen Lusin's theorem

Let $f$ be a measurable function on a subset $E$ of $\mathbb{R}^n$. Lusin's theorem states that for any $\epsilon>0$, there exists a measurable subset $F$ such that $F$ open in $E$, ...
0
votes
1answer
69 views

Why is $fg$ integrable w.r.t. a probability measure if $f,g$ are Lebesgue integrable?

In one of the proofs, my text mentions that if $f,g$ are Lebesgue integrable then $fg$ is integrable with respect to a probability measure. I guess I have missed something, since it doesn't look ...
1
vote
2answers
71 views

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions?

Is there an analogue of Lebesgue’s Dominated Convergence Theorem for a net $ (f_{\alpha})_{\alpha \in A} $ of measurable functions defined on a measure space $ (\Omega,\Sigma,\mu) $, where the index ...
5
votes
0answers
58 views

An inequality between integrals of series of characteristic functions of cubes

Let $1\leq p<\infty$. Prove that there exists $C>0$ such that $$ \left(\int\left|\sum_{i=1}^\infty a_i\chi_{2Q_i}\right|^p \, dx\right)^{1/p} \leq C\left(\int\left|\sum_{i=1}^\infty ...
1
vote
1answer
21 views

Probably simple application of Holder or Minkowski for integrals

This is a step in a lecture note I'm reading. It should be simple because the author considers it obvious but I can't see it. What am I missing? Suppose $U$ and $V$ are integrable over measure space ...
1
vote
0answers
20 views

Change of variable for Lebesegue Integral

Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds ...
1
vote
0answers
50 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
2
votes
1answer
18 views

Fine Print of Egorov's Theorem

The Egorov's theorem in my textbook requires that the function to be define on a set with finite measure. Why is this necessary, please? Thank you!
0
votes
1answer
40 views

Two questions on Fatou's Lemma

While reading the following paragraph from Real Analysis by Stein (I hope this does not breach any copyright; if so, I have to type it out), two questions occurred to me. In the proof of Fauto's ...
1
vote
1answer
34 views

$L^p$ norm of a measurable function is bounded by its operation on step functions

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$ $$ \left|\int_0^1 fg d\mu\right|\leq \|g\|_q. $$ Prove that ...
2
votes
1answer
46 views

a condition given by step functions implies the condition holds for L^q space

Let $1\leq p<\infty$, $1/p+1/q=1$. Let $f$ be a measurable function on $[0,1]$ such that for all step functions $g$ on $[0,1]$, $$ |\int_0^1 fg d\mu|\leq ||g||_q. $$ Prove $||f||_p\leq 1$. How ...
1
vote
1answer
15 views

comparing mean values of a positive functions

Suppose that $D$ is a bounded open set in $\mathbb{R}^{n}$, and $A\subset B\subset D$ measurable sets. Let $f:D\rightarrow [0.+\infty)$ be a measurable function (or even locally integrable) and ...
0
votes
1answer
29 views

Integration with respect to conditional measure?

Let $(X_n)$ be a Markov chain. For $i\in S$ my text defines $$N_i:=\sum_{n=0}^\infty \mathbf 1_{\{ X_n=i \}}$$ and then, as a part of a larger proof, claims that $$\mathbb E_i(N_i)=\sum_{n=0}^\infty ...
1
vote
1answer
37 views

Limit of average of real function

I need some hints regarding this exercice. if $f : [0, \infty)\rightarrow \mathbb{R}$ is a measurable function s.t $\lim_{x\rightarrow \infty} f(x) = a$, prove : \begin{align} \lim_{x\rightarrow ...
2
votes
1answer
72 views

Spivak's “Calculus in Manifolds” problems

I have some troubles with this problems. Problem 1.18: If $A \subset [0,1]$ is the union of open intervals $(a_i,b_i)$ such that every rational number of $(0,1)$ is contained in $(a_i,b_i)$, for ...
0
votes
2answers
23 views

Exercise on abstract integration

Let $f_n$ be a sequence of nonnegative functions defined on $\mathbb{R}^N$ such that $f_n \rightarrow f $ almost everywhere on $\mathbb{R}^N$ and such that $$\int_{\mathbb{R}^N} f_n \rightarrow ...
0
votes
1answer
24 views

the area of the image under a specific holomorphic function of the unit disk

Let $f(z)=z^3+\frac{z^2}{2}$. Let $D$ be the unit disk in $\mathbb{C}$. How to compute $$ Area(f(D))? $$ In the case that $f:D\to \mathbb{C}$ is injective, \begin{align*} Area(f(D))&= \int_D ...
2
votes
3answers
43 views

Question on integral, notation and Nikodym derivative

I have a very general question for those measure theoric, real analysis guy out there . I am very confused by the concept of Nikodym derivative. If $v << \mu$, we can find a non negative ...
2
votes
1answer
36 views

Reversing limits in Lebesgue integration

I know that reversing limits of Riemann integration is possible by putting minus sign. My question is that there is a similar result for Lebesgue integral as well. For example, $$ ...
2
votes
2answers
66 views

solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$ |f(t,x)|\leq \phi(t)|x|. $$ Suppose ...
1
vote
1answer
33 views

inequalities concerning integration and measure

Let $f$ be a non-negative function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} f=1$. Let $p\in(0,1)$. Let $E$ be any measurable subset of $\mathbb{R}^n$. Prove that $$ \int _E f^p\leq ...