Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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0
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2answers
151 views

If $\{f_{i}\}_{i\in I}$ are continuous functions $f_{i}:\, X\to\mathbb{R}$ then ${\displaystyle \sup_{i\in I}f_{i}}$ is measurable?

I saw the following exercise: Prove or give a counterexample: If $\{f_{i}\}_{i\in I}$ are continuous functions $f_{i}:\, X\to\mathbb{R}$ then ${\displaystyle \sup_{i\in\mathbb{I}}f_{i}}$ is ...
3
votes
1answer
257 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
1
vote
1answer
56 views

Proving that if $\{f_{i}\}_{i\in\mathbb{N}}$ is a sequence of measurable functions then so is $\displaystyle\sup_{i\in\mathbb{N}}f_{i}$.

I wish to prove that if $\{f_{i}\}_{i\in\mathbb{N}}$ is a sequence of measurable functions then so is $\sup_{i\in\mathbb{N}}f_{i}$. From another question I asked today I know that it suffices to ...
4
votes
1answer
192 views

Finite additivity, atomlessness and countable additivity

So, I'm trying to get my head around when you can have finitely but not countably additive probabilities. The standard example of a finitely additive but not countably additive space is the ...
3
votes
1answer
126 views

Proving a sufficient and necessary condition for $f:\, X\to\mathbb{R}\cup\{\pm\infty\}$ to be measurable

I saw the following question: Denote $\overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}$, the open sets containing $x\in\mathbb{R}$ are the open sets in $\mathbb{R}$ containing $x$. The open ...
1
vote
1answer
709 views

Change of variables in integration

Under certain conditions on the functions $g:\mathbb{R}^n \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ involved I have seen formulas such as $$ \int \limits _{\mathbb{R}^n} f\circ g (x) \,dx = ...
2
votes
1answer
102 views

Three-Dimensional Lebesgue Measure

I've been trying to calculate the three-dimensional Lebesgue measure of $$\left\{(x,y,\theta)\in\mathbb{R}^2\times[0,\pi):\ x^2+y^2\leq 1;\; \theta\in[0,\pi);\; ...
1
vote
1answer
163 views

absolute continuity in trigonometric functions

I want to show the function $f:[0,1]\to \mathbb{R}$ $$f=\left\{ \begin{array}{ll} x^{3/2}\sin\left(\frac{1}{x}\right), & {x \in (0,1]} \\ 0, & x=0 \end{array} \right.$$ is ...
4
votes
1answer
66 views

Using the SLLN to show that the Sample Mean of Arrivals tends to the Arrival Rate for a simple Poisson Process

Let $N_t = N([0,t])$ denote a Poisson process with rate $\lambda = 1$ on the interval $[0,1]$. I am wondering how I can use the Law of Large Numbers to formally argue that: $$\frac{N_t}{t} ...
1
vote
0answers
48 views

Variation of a refinement

Let $P$ be a partition of $[a, b]$ that is a refinement of the partition $P'$. For a real-valued function $f$ on $[a, b]$, show that $V(f, P') \leq V(f, P)$. Proof: Let $P'$= {$a$ = $x_{0}$ < ...
0
votes
1answer
110 views

Is this function Measurable?

Suppose $f$ is a continuous function defined on $[0,1]$. Let $\operatorname{sgn}(f)$ denote the signal function of $f$. Is $\operatorname{sgn}(f)$ a measurable function?
-1
votes
1answer
143 views

Calculating the Lebesgue decomposition of a measure

How we should calculate the Lebesgue decomposition of a measure? Please explain it with an example such I can get the whole idea behind it.
2
votes
1answer
439 views

How to evaluate integrals with respect to Lebesgue measure on the unit sphere?

Let $\sigma$ be the "normalized Lebesgue" (Haar, really...) measure on the unit sphere $S=S^{n-1} \subset \mathbb R^n$. That is, $\sigma$ has support $S$, it is uniformly distributed, and $\int_S ...
0
votes
1answer
77 views

How to prove this problem about integrable function?

If an integrable function $f(x)\ge0$ a.e., then $\int fd\mu\ge0$. Any hint is appreciated.
3
votes
2answers
163 views

Borel sets including $\infty$

Can someone give me a hint how to show the equality $\{ B \cup F \mid B\in \sigma(\{(-\infty,a) \mid a\in \mathbb{R}\}),\ F\subseteq\{-\infty,+\infty\} \}=\sigma(\{ [-\infty,a) \mid a \in \mathbb{R} ...
0
votes
1answer
93 views

Question in product measure

How can I prove the product of two measurable functions in the product measure space is measurable? I tried but still do not know how to prove.
2
votes
1answer
128 views

$F_\sigma$ subset of equal outer measure

Let $E\subset[a,b]$. How can we show that: $\exists F \in F_\sigma$: $F\subseteq E$ and $\mu^*(F)=\mu^*(E)$ Can we conclude that $E$ is measurable?
0
votes
0answers
111 views

How to do Lebesgue decomposition in this problem?

$\{x_n\}$ is a sequence in $\mathbb{R}$ and $\{p_n\}$ is a sequence of positive numbers. Define a $\sigma$-finite measure $\nu(E)=\sum_{x_n\in E}p_n$. Find the Lebesgue decomposition of $\nu$ with ...
1
vote
1answer
97 views

On one property of the Lebesgue Measure

Does there exist a set $E\subset [0,1]$ with $m(E)<1$ such that $m(E\cap I)\geq m(I)/2$ for all measurable sets $I\subset [0,1]$? I am not able to construct one, but it seems possible. Any help ...
3
votes
4answers
89 views

Formally proving that if $E[\sum_{i=1}^\infty {X_i}] < \infty$, then $\sum_{i=1}^\infty {X_i} < \infty$ almost surely

Given a sequence of non-negative random variables $(X_i)_{i\in\mathbb{N}}$, I would like to show that $$ \mathbb{E}[\sum_{i=1}^\infty {X_i}] < \infty$$ implies that $$ \sum_{i=1}^\infty {X_i} ...
0
votes
2answers
411 views

Fundamental Theorem of Calculus for Lebesgue Integral

I am trying to prove (96) on pg. 324 of baby Rudin. ie. That is $f$ is Riemann integrable on $[a,b]$ and if $$F(x)=\int_a ^x f(t)dt$$ then $F'=f(x)$ $a.e$ on $[a,b]$ Anything I do seems to just ...
4
votes
2answers
73 views

On an identity about integrals

Suppose you have two finite Borel measures $\mu$ and $\nu$ on $(0,\infty)$. I would like to show that there exists a finite Borel measure $\omega$ such that $$\int_0^{\infty} f(z) d\omega(z) = ...
1
vote
0answers
489 views

Vitali Covering

Let $E$ be a set of finite outer measure and $F$ a collection of closed, bounded intervals that cover $E$ in the sense of Vitali. Show that there is a countable disjoint collection ...
9
votes
5answers
1k views

Simpler proof - Non atomic measures

Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)>0$ there exists $F \subset E$ measurable such that $0<\mu(F) ...
1
vote
0answers
65 views

Counter-example to homogeneity of unsigned lower Lebesgue integral.

I know this is sort of a nonissue, but in one of the exercises the author asks us to prove If $0\leq c<+\infty$, then $\int cf=c\int f$ where $f:\mathbb{R}^{d}\to[0,+\infty]$ (note $f$ is not ...
2
votes
0answers
554 views

Show that $f$ is of Bounded Variation by $f'$' being integrable.

For $\alpha$, $\beta$ > 0, define the function $f$ on $[0, 1]$ by $f(x) = x^{\alpha}\sin(1/x^{\beta})$ for $0 < x \leq 1$ and $f(x) = 0$ for $x=0$. Show that if $\alpha > \beta$, then $f$ is of ...
4
votes
2answers
133 views

Is the set of measures of all measurable sets with finite measure closed?

Let $(X,M,\mu)$ be a complete measure space. Does the set $\{\mu(E)|E\in M ,\mu(E)<\infty\}$ have to be a closed subset of $R$ ? Thank you
5
votes
1answer
170 views

Properties of certain type of measures

Let $p\in (0,1)$. For each $k\in\mathbb{N}$ and tuple $(\varepsilon_1,\ldots,\varepsilon_k)\in\{0,1\}^k$ denote $$ S_{\varepsilon_1,\ldots,\varepsilon_k}=\left\{\sum\limits_{j=1}^\infty x_j 2^{-j}: ...
4
votes
1answer
212 views

Example of a sequence $f_n \in L^1(\mathbb{R})$ with $f_n \to f$ uniformly, but such that $f \not \in L^1(\mathbb{R})$ and $\int f_n \not \to \int f$

For practice, I'm working through some of the exercises in Folland's "Real Analysis: Modern Techinques and Their Applications." In Chapter 2, Exercise 19, Folland asks for sequences of functions $f_n ...
0
votes
1answer
72 views

If f is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function g does that imply that f=0 a.e

If $f$ is an $L^p$ function and $\int f(x)g(x)dx=0$ for every $L^p$ function $g$ does that imply that $f=0$ a.e
0
votes
1answer
194 views

convergence in $L^2$

Let $ (f_n)$ be a sequence of measurable functions on $ [0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$. Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty ...
0
votes
0answers
115 views

Explanation of the Girsanov's transformation

The Girsanov's theorem is making me all confused. In my course literature they explain it by some simple discrete examples of coin-tossing etc. Saying that $Z$ is the ratio of $\frac{P^a(A)}{P(A)}$ ...
2
votes
2answers
193 views

The limit of the Lebesgue integration

If $E\subset R^n$ be a Lebesgue measurable set and let $(f_k)$ be a sequence of non-negative Lebesgue measurable function on E such that $\lim f_k=f$ a.e. I want to prove that if $\int_E ...
1
vote
1answer
159 views

Measurability of a function on a product measure.

I was hoping someone can point me in the right direction for the proof of this question. I have some idea of whats going on, but I need a little more. Anyway, here is the statement. We are given that ...
1
vote
1answer
52 views

Upper and Lower Differentiability

Show that if $f$ is defined on $(a, b)$ and $c\in(a, b)$ is a local minimizer for f, then $\underline{D}f(c) \leq 0 \leq \overline{D}f(c)$. Proof: There exists $\delta > 0$ such that $f(c) < ...
1
vote
1answer
78 views

The continuous measure function

If $f$ is a Lebesgue measurable function on $R^n$, we define $$K(t)=\lambda\{x\in R^n:|f(x)|>t\}.$$ I want to prove that $\int_0^{\infty}K(t)dt=\int_{R^n} |f(x)|dx$ If $ f\in L^1 (R^n)$, then ...
2
votes
2answers
297 views

Sigma-algebras generated by maps

Let $(\Omega,\mathbb{F},P)$ be a probability space and $\epsilon_1,...,\epsilon_n$ be real-valued random variables defined on $\Omega$. Now let $\mathbb{D}_n$ be the sigma-algebra generated by ...
2
votes
1answer
69 views

Is there an almost continuous function which cannot be almost equal to any other cont.function?

Is there an almost continuous function which cannot be almost equal to any other continuous function?
6
votes
2answers
209 views

A Borel algebra containing $\infty$

If I adjoint $\infty$ to the real numbers ($\overline{\mathbb{R}}=\mathbb{R}\cup\infty$) is there a reasonable way to define a $\sigma$-algebra "$\mathcal{B}_{\overline{\mathbb{R}}}$" such that ...
2
votes
1answer
246 views

Polynomials are dense in $L^{3/2}([0,1])$

Show that if $f\in L^{3/2}([0,1])$ satisfies $$ \int_0^1x^nf(x)dx = 0,\:\: n\in\mathbb{N} \cup \{0\}$$ then $f = 0$ a.e. I have not taken a course in integration theory, but my guess is that this ...
2
votes
1answer
387 views

How to prove it is not a $\sigma$-field?

$X=(-\infty,\infty)$, $\mathcal{F}_n$ is the $\sigma$-field generated by $[0,1),[1,2),...,[n-1,n)$. Prove $\mathcal{F}_n\subset \mathcal{F}_{n+1}$ and $\cup_{n=1}^\infty\mathcal{F}_n$ is not a ...
3
votes
1answer
257 views

Nice applications of the Haar measure

The existence of the Haar measure is a beautiful result that has a lot of applications. For example, one can prove using the Haar measure that the category of representations of a compact Lie group is ...
4
votes
1answer
135 views

Volume is a Continuous Function

I am working on the following problem: Suppose $C \subset \mathbb{R}^d$ is a compact and non-empty set. Let $C_0 = C$ and let $C_t = \{x \in \mathbb{R}^d : d(x,C) \leq t \}$ for all $t >0$. ...
3
votes
1answer
566 views

Variant of the Vitali Covering Lemma

I am working on the following problem, which is based on a problem from Stein and Shakarchi: Prove the following variant of the Vitali Covering Lemma: If E is a set of finite Lebesgue measure in ...
5
votes
2answers
134 views

Derivative of $t \mapsto \Vert f+tg \Vert_p^p$

Suppose $(X,\mathcal A, \mu)$ is a measure space and let $f,g\in L^p(X)$ be real-valued functions, $p\in(1,+\infty)$. Let us define $$ F:\mathbb{R} \ni t \mapsto \int_X \vert f(x)+tg(x) \vert^p ...
2
votes
1answer
280 views

Counter example for $F_n \downarrow F$ then $\int f_n d\mu \downarrow \int f d\mu$?

Is there any counter example for for $F_n \downarrow F$ then $\int F_n \, d\mu \downarrow \int F \, d\mu$? I came up with the one below but $F_n$ does not go down to $0$ monotonically. I need ...
11
votes
3answers
424 views

Probability of selecting a non-measurable set

If you randomly select a subset of $[0,1]$, what is the probability that it will be measurable? Edit: This question may be unanswerable as asked. If additional assumptions could be made to make it ...
4
votes
1answer
2k views

Are Monotone functions Borel Measurable?

Could you guide me how to prove that any monotone function from $R\rightarrow R$ is Borel measurable? Should we separate the functions into continuous and non-continuous? How to prove for not ...
0
votes
1answer
68 views

How can I show this inequality?

Let $\lambda B=\{x\in\mathbb{R}^n:\ \|x\|<\lambda\}$. Let $\eta>0$, $r_n\in (0,\eta)$ and $r_n\rightarrow \eta$. Suppose $u$ is a measurable function defined in $\eta B$. How can i show that ...
4
votes
2answers
93 views

Measure Inequality

Let $\lbrace I_1,\ldots I_k \rbrace$ be a collection of bounded intervals. Choose $I_1$ to be of the largest. Denote $T=\lbrace i\in \lbrace 1,\ldots ,k\rbrace \mid (I_1 \cap I_i)\not= ...