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

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2
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1answer
42 views

Sigma algebra on space of signed Radon measures

consider the space $M = \left\{ \mu : \mathscr{B}(\mathbb{R}) \to \mathbb{R} \cup \left\{ -\infty, +\infty \right\} \ | \ \mu \text{ signed Radon measure} \right\}$ which is not a vector space, since ...
0
votes
1answer
18 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
14
votes
0answers
323 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 ...
2
votes
1answer
66 views

Does every nonmeasurable set split into a measurable subset and a purely nonmeasurable subset?

Being curious I'm wondering: Suppose you're given a continuous function over a Borel space. Then the preimage of every open is measurable. However, while the preimage of every neighborhood of some ...
1
vote
1answer
38 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
votes
1answer
24 views

Cohn measure theory -page 17

I can't understand the proof of equation $(2)$ of the theorem $1.3.6$ (page $17$), "As to the induction step, note that the $\mu$-measurability of $B_{n+1}$ and the disjointness of the sequence $B_i$ ...
1
vote
1answer
105 views

Real valued random variables and cumulative distribution functions (c.d.f.)

Let $X$ be a random variable with values in $\mathbb R$ (we fix the Lebesgue measure on $\mathbb R$), then is well defined a c.d.f. $F_X$ such that $$F_X(x)=X_\ast P(]-\infty,x])=P(X\in]-\infty,x])$$ ...
0
votes
0answers
31 views

Halmos Measure Theory section 19 Theorem C

I have trouble explaining the claim "the measurability of f+g follows from Theorem A". If I can show that $N(f+g)=\{x:(f+g)(x)\neq0\}$ is measurable if f and g are measurable then the proof is ...
1
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1answer
52 views

Definition of the total variation of a measure: countable partitions versus finite partitions

The total variation according to Rudin is defined as: $$|\mu|(E):=\sup_{\bigcup_{k\in\mathbb{N}}E_k=E}\sum_k|\mu(E_k)|$$ where the supremum is taken over all countable partitions. Now I'm reading in ...
0
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0answers
26 views

The projective limit of probability spaces and the Kolmogorov-Daniell theorem

Does the "projective limit" concept exist for probability spaces? The only result that I know of seems to be the Kolmogorov-Daniell theorem, but this is just a particular case where the spaces ...
0
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1answer
31 views

Calculate Radon-Nikodym derivative in a point when it is continuous in that point

I can't solve the following exercise, even if I find it quite intuitive. Let $\nu, \mu$ be Radon measures on a metric space $(X,d)$. Suppose that: 1) $w\in L^1(X,\mu), w\geq 0$ $\mu$ a.e.; 2) $w$ is ...
0
votes
1answer
42 views

An example of stochastic process

I use the following definition for a stochastic process. Let $(\Omega, \mathcal F, P)$ be a probability space, $(E, \mathcal E)$ be a measurable space, and $T$ be a non-empty set. A collection ...
1
vote
1answer
32 views

$F_n \overset{w}{\to} F$, and $F$ is continous. Show that $F_n$ converges to $F$ uniformly on $\mathbb{R}$

$\{F_n\}$ and $F$ are distribution functions, and $F$ is continuous on $\mathbb{R}$. If $F_n$ converges weakly to $F$, show that $$ \sup_x | F_n(x) - F(x) | \to 0, n \to \infty $$ I know that ...
0
votes
1answer
11 views

can representatives of an equivalence class of L_1(r^d)be nonmeasurable

Motivation: The composition of 2 Lebesgue measurable functions need not be measurable. This problem can be dealt with in a case by case basis( like with convolutions). Or as Big Rudin does, apply ...
1
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1answer
93 views

Measure Theory Book

What book should I use for measure theory?I have solved Rudin's Principle Of mathematical analysis up to chapter 7.Some people advised me to use Real and complex analysis by Rudin, while other said it ...
0
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0answers
26 views

Sigma-additive measure on algebra of sets

Most of the time, $\sigma$-additive functions are defined on $\sigma$-algebras of sets. Is it possible to define a $\sigma$-additive function on an algebra of sets? If so, what is the ...
0
votes
1answer
36 views

Dense subset of $[0,1]$ with Lebesgue measure $\epsilon$

We wish to find a Lebesgue measurable subset of $[0,1]$ that is in dense in $[0,1]$ with measure exactly $\epsilon$, where $\epsilon \in (0,1)$. My idea is to let $I=(0,\epsilon)$ and let ...
0
votes
0answers
40 views

classifying topological spaces by measures

While looking at some spaces, I happend to know,that in some spaces(like $\mathbb R^n$) Null sets have topological properties(defining the Algebra by the open sets)! some examples: in $\mathbb R^n$ a ...
2
votes
2answers
51 views

Banach spaces not isomorphic to $\ell^p(S)$?

We know that every Hilbert space is unitarily equivalent to $\ell^2(S)$, for a set $S$ of suitable cardinality. Is there a Banach space which is NOT isomorphic to $L^p(X)$, for any $1\leq p \leq ...
1
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1answer
69 views

A Question on Sets of Full Outer Measure

I came across this problem whilst studying for a comprehensive exam in real analysis; for reference, see Exercise 1.24(A) in Folland's Real Analysis; it's a modification of that. Consider the unit ...
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0answers
36 views

When is an outer Borel regular measure (inner and outer) regular?

Let $X$ be a topological space and $\mu$ an "outer" Borel regular measure on $X$ (for all $A\subset X$, there is $B$ Borel with $\mu(A)=\mu(B)$). Assume that $X=\cup _{i=1}^\infty U_i$, where each ...
1
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0answers
31 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
3
votes
1answer
80 views

Every Lipschitz function is the primitive of a measurable function

I was doing exercise 5 of this exercise sheet and I don't know how to conclude. I need to prove that if $f \colon [0,1]\to \mathbb{R}$ is Lipshitz, $X$ is a uniform$(0,1)$ random variable and ...
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0answers
11 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
1
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0answers
15 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
0answers
10 views

Two Radon measures and mutual singularity

Let $\mu$ and $\lambda$ be Radon measures on $\mathbb{R^n}$. Show that $\mu$ and $\lambda$ are mutually singular iff $D(\mu,\lambda,x)=\infty$ for $\mu$ almost all $x \in \mathbb{R^n}$. I have looked ...
2
votes
2answers
46 views

Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?

A measure on $\mathbb R$ is a set function $\mu,$ defined for all Borel sets of $\mathbb R,$ which is countably additive(that is, $\mu(E)=\sum \mu(E_{i})$ if $E$ is the union of the countable family ...
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0answers
86 views

Riemann implies Lebesgue integrablility on $\mathbb{R}^n$, prove $f(x)$ continuous at x where $g(x)=G(x)$

Let $f:[a_1,b_1]\times \cdots \times[a_n,b_n] \rightarrow \mathbb{R}$ be Riemann integrable. Prove that is $f$ Lebesgue integrable. Proof: $$Q:= [a_1,b_1]\times \cdots \times [a_n,b_n].$$ For simple ...
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0answers
26 views

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

$f:[a_1,b_1]x[a_2,b_2]\rightarrow \mathbb{R}$ 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
0answers
35 views

Integrable function $f$ and simple function $\phi$ such that $ \int{|f-\phi|} \> d\mu < \epsilon.$

I am seeking solution verification for the following problem. Suppose $f$ is an integrable function. We wish to show that there exists a simple function $\phi$ such that $$ \int{|f-\phi|} \> d\mu ...
0
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1answer
34 views

Convergence in measure theory

Let $u_n$ be a sequence converging to $u$ in $L^2$. Let $f(t)$ be a bounded continuous function. Can I say that $f(u_n)$ converges to $f(u)$ in $L^p$ for every $1<p<\infty$?
1
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0answers
33 views

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. [duplicate]

Show that the series $\sum\limits_n P(A \cap A_n)$ converges for $A=\bigcap\limits_{n=M}^{\infty} A_n^{c}$. From here http://math.stackexchange.com/a/878635/140308 (proof attempt is there too) Sorry ...
0
votes
1answer
26 views

A field being a sigma field if and only if it's a monotone class

The exercise is as follows: "The limit of an increasing (or decreasing) sequence An of sets is defined as its union ∪nAn (or the intersection ∩nAn). A monotone class is defined as a class ...
0
votes
0answers
11 views

Is every frame homomorphism induced by a measurable function?

Let $M$ be the Lebesgue measure algebra of the unit interval $[0,1]$, i.e. equivalence classes of Lebesgue measurable sets modulo sets of measure $0$. This is a complete Boolean algebra, hence in ...
0
votes
1answer
36 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 ...
1
vote
3answers
52 views

Set of points at which sequence of measurable functions converge (another approach)

Question is to prove that : Set of all points at which a sequence of measurable functions converge is a measurable set.. What i have tried is as follows : We are looking at the following set : ...
0
votes
1answer
26 views

Vitali set of outer measure 1

How to construct a Vitali set of outer measure 1. I couldn't understand the argument given here. Isn't there any easier way? I would also like if someone explains that to me. Thank you in advance!
0
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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 ...
-1
votes
1answer
39 views

Counterexamples in measure theory

Can you suggest me a book which primarily deals with counter-examples in measure theory? Thank You in advance!
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0answers
24 views

Borel measure which is not regular

I need an example where a Borel measure is not regular. I already proved that any finite Borel measure is regular and tight. The examples of irregular measures given here are too rigid and not so ...
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
2answers
25 views

Totally disconnected measurable set with positive measure

Could we find a totally disconnected set of the real numbers which is Lebesgue measurable and has positive measure?
1
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1answer
30 views

$\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots, S_n)$ generated by partial sums

is true that $\sigma(X_1,\dots, X_n) = \sigma(S_1,\dots,S_n)$ where $S_n=\sum_{i=1}^n X_i$ in general or I have to impose additional restrictions to the random variables (for instance, independence)? ...
4
votes
2answers
126 views

The set $E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\}$ does not contain all irrational numbers in $[0,1]$

Let $q_1,q_2,q_3,...$ be an enumeration of $\mathbb{Q}\cap[0,1]$ and let $r,t \in (0,1).$ Consider the set $$E= \{x\in [0,1]: \sum_{j=1}^\infty t^j|x−q_j|^{-r} <\infty\} $$ (a) Show that $E\neq ...
2
votes
0answers
34 views

Asymmetry in definition of regular measure

In a Borel measure space $(X, \mathcal{B}, \mu)$, $\mu$ is outer regular at $E$ if \begin{equation} \mu(E) = \inf_{U \textrm{ open}} \{\mu(U): U \supseteq E\} \end{equation} and ...
3
votes
2answers
80 views

Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
0
votes
2answers
68 views

Uniform integrability of a function in $L^1$

A collection of functions $(\phi_i)_{i\in I}\in L^1(\mu)$ is called uniformly integrable if given $\epsilon>0$ there exists $\delta>0$ such that : $$\int_E|\phi_i|d\mu<\epsilon~~~~\forall ...
2
votes
1answer
44 views

Why $m(B^n(0, r)) = c_nr^n$?

(Bear with me: I realize this is quite basic question, but I'm a little loss at how to search for an answer). Anyway: I came across a real analysis proof which uses a property that (as far as I can ...