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

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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 ...
1
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1answer
55 views

Why define the Lebesgue-Integral just for measurable functions?

Usually, the Lebesgue integral, for example on Wikipedia, is defined for non-negative measureable functions as $$ \int_E f \, d\mu := \sup\left\{ \int_E s \, d\mu : 0 \le s \le f, s \text{ simple } ...
2
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1answer
70 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
2
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1answer
46 views

$f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$?

I am trying to show that the sequence of functions $f_n(x) = x(1-x)^n$ converges uniformly to $0$ on $[0, 1]$. Well at $0$ and $1$, $f_n(x) = 0$ for all $n$. So let $x \in (0, 1)$. $f_n(x)$ ...
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2answers
32 views

Jordan Content of the set $\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, … \}$?

I have the following definition of the Jordan Content of set $E$ - inf $\{ \sum_{i=1}^n |I_i| : n \in \mathbb{N}, I_1, I_2, ..., I_n$ intervals such that $E \subseteq \bigcup_{i=1}^n\}$ That seems ...
1
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1answer
17 views

Coin tossing, two heads always followed by two tails - lim sup necessary?

In Bernoulli Space $\Omega$, let $E_n$ be the event that the $n$th toss is heads. Write down a formula in terms of the $E_n$ for the following event: “Every time two Heads appear in succession, the ...
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3answers
33 views

How many algebras of subsets of $X$ contain exactly four elements?

Let X be a set with five elements. How many algebras of subsets of X contain exactly four subsets? Well $\emptyset, X$ must be in any algebra of subsets of $X$ so that means we have to find two more ...
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1answer
11 views

Outer Measure is not Finite Additive

A cube $Q$ in $\mathbb{R}^d$ is a subset $[a_1,b_1]\times \cdots \times [a_d,b_d]$ of $\mathbb{R}^d$, where $a_i,b_i\in\mathbb{R}$ and $b_i-a_i=b_j-a_j$ for all $i,j$. By volume of cube $Q$, denoted ...
0
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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 $ ...
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0answers
31 views

Invariant Measure

Let $\dot{x}=u(x)$ a dynamical system ($x\in\Gamma$) with solution $x(t)=\Phi^t_u(y)$ and $\mu$ a $\Phi^t_u$-invariant measure on $\sigma_\Gamma$. I want to show that the smooth density $\rho=d\mu/dV$ ...
2
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1answer
24 views

Absolutely continuous probability measures example

I was given the following definition: Then this example: It is said that $\mathbb P_1\ll\mathbb P_2$ , but I don't really see it.Please help.
2
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2answers
62 views

From distribution to Measure [duplicate]

I have been asked to create a new post with my question. So it is about starting from a distribution function and proving that we can always find a probability space. My attempt is this : So assume ...
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0answers
28 views

Relation between $L^1(T)$ and $L^1[0,1]$

I know the question may be too general, but I need to know if there is a way in which I could relate the spaces $L^1(T)$ (where $T=\{e^{2 \pi i x}: x \in [0,1]\}$ and we use the Lebesgue measure on ...
1
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0answers
29 views

From distribution function to probability measure

So assume we have a probability space $(\Omega, \mathcal{F}, P)$ and a random variable $X : \Omega \rightarrow \mathbb{R}^*$. We can derive from this a distribution $P_X$, and a distribution function ...
1
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1answer
32 views

Measurability of a Borel function

I need some help on the following proof. The claim is: Suppose $f:\mathbb{R}^k \to \mathbb{R}$ and $f \in B(\mathbb{R}^k)/B(\mathbb{R})$. i.e. Borel measurable. Let $X_1$,...,$X_k$ be random ...
1
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1answer
59 views

How do you show $\int \limits_{X \times Y} f(x,y)\, d\lambda < \infty$ if $\int \limits_{X} \int \limits_{Y} f(x,y) \,d\nu \,d\mu < \infty$?

Suppose $f: X\times Y \rightarrow [0,\infty]$ is a measurable function with respect to the product measure $\lambda$ ( $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces). Suppose ...
1
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0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
1
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1answer
34 views

Jordan Measure and Lebesgue Measure

The Jordan outer measure of $J^*(E)$ a set $E\subseteq \mathbb{R}$ is defined as infimim of $\sum_{i=1}^n (b_i-a_i)$ where $(a_i,b_i)$ are open intervals whose union contains $E$. The Jordan inner ...
1
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2answers
69 views

Measure of set of rational numbers

I find it difficult to understand why the 'size' of the set of rational numbers in an interval such as [0,1] is zero. I know that there are way more irrational numbers than rational numbers such that ...
0
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1answer
45 views

What is the Lebesgue measure of a following set?

This might be trivial for some of you. Let $E$ be a set defined by $ E= \{(x,y): a<\frac{x}{y}<b, c<\frac{y}{x}<d \}$. What is the Lebesgue measure of this set. Measure should be an ...
3
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2answers
54 views

Convergence in measure and $L_p$ implies product converges in $L_p$

This was given on an old comp as a true or false problem: If $1<p<\infty$, $|f_n|\leq 1$, $f_n\rightarrow f$ in measure, and $g_n\rightarrow g$ in $L_p$, then $f_ng_n\rightarrow fg$ in $L_p$. ...
2
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1answer
39 views

Measure and additivity

I am learning Measure Theory on my own, so please forgive me if my question is naive. Finitely additive and $\sigma$-additive measures are defined in a natural way on finite algebras and ...
2
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0answers
45 views

Is a Lebesgue measurable subset a null set if each compact subset is a nullset?

If $A \subset \Bbb{R}$ is Lebesgue measurable.($m$ is Lebesgue measure) and for every compact $K \subset A$, $m(K)=0$; is it true that $m(A)=0$?
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1answer
32 views

Essential support vs. classical support for a continuous function

The essential support of a function $f:\Bbb R^n\rightarrow \Bbb R$ is defined in the following way: Let's denote $\mathcal A_f=\{\omega \subset \Bbb R^n: \omega \quad \text{open}, \quad f(x)=0\quad ...
2
votes
2answers
32 views

Existence of an extending measure

Let $\Omega$ be a nonempty set and $\cal{A}$ be any class of subsets of $\Omega$ including emptyset. Suppose that $\mu:\cal{A}\to R^{+}\cup{+\infty}$ be such a non-zero function that the equality ...
1
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0answers
16 views

Measures on $\mathbb{P}^1(\mathbb{C})$ and $\mathbb{P}^1(\mathbb{C}_p)$?

Is there a natural measure on the above spaces? Ideally, I would like a measure that is invariant under automorphisms of $\mathbb{P}^1$.
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1answer
17 views

From the given measure $\mu,$ how to construct another measure $\mu^{\ast}$; so that $d\mu^{\ast}(y)= (1+y^{2})d\mu(y)$?

Put $\mu= \sum_{n\in \mathbb Z}c_{n}\delta_{n};$ where $\delta_{n}$ is the unit Dirac mass at $n.$ We note that, $\mu$ is a complex Borel measure on $\mathbb R$ and the total variation of $\mu,$ that ...
3
votes
1answer
29 views

The existence of conditional expectation with respect to a sub-$\sigma$-algebra

I was trying to solve the exercise 3.17 from the book of real analysis by Folland and I've found a problem. The first part of the exercise is the following: Let $(X, M, \mu) $ be a $\sigma$-finite ...
1
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1answer
59 views

Let F be a distribution function. Prove that X is a RV.

Let F be a distribution function. On $(\Omega, \mathfrak{F}, P)=((0,1), \mathfrak{B}(0,1),\lambda)$ where $\lambda$ denotes Lebesgue measure. Define X: $\Omega \ to \mathbb{R}$ by $X(\omega) = sup(y ...
3
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0answers
32 views

Can we expect, $h\ast \mu \in L^{2}(\mathbb R, (1+|x|^{2})^{s})$ for $h\in \mathcal{S}(\mathbb R), \mu\in M(\mathbb R)$ and $s>1/2$?

We put, $M(\mathbb R)=$ The space of complex bounded Borel measure on $\mathbb R$ [With each complex Borel measure $\mu$ on $\mathbb R$ there is associated a set function $|\mu|,$ the total variation ...
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2answers
63 views

If a probability space has no measurable subsets with $P$ strictly between $0$ and $1$, then every random variable is constant a.s.

Let $(\Omega, \mathfrak{F}, P)$ be a probability space such that $\forall F \in \mathfrak{F}, P(F) = 0 \ or \ 1$. Show that for all random variables X on $(\Omega, \mathfrak{F}, P)$, $\exists \ c ...
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2answers
58 views

If X and Y are random variables with the same distribution, prove that f(X) and f(Y) are random variables that have the same distribution.

Suppose X is a RV on $(\Omega, \mathfrak{F}, P)$. Let f be Borel-measurable on $(\mathbb{R}, \mathfrak{B})$. 1 Show that f(X) is also a RV on $(\Omega, \mathfrak{F}, P)$. 2 Let Y ba RV on ...
1
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1answer
36 views

$\liminf(x_{n}+y_{n})= x + \liminf(y_{n})$ if $x_{n} \rightarrow x$.

As I'm studying measure theory, I've seen the fact in the title of my post used several times. In particular, it has been used in the proof of Lebesgue's Dominated Convergence Theorem. (I, myself, ...
4
votes
2answers
100 views

Fubini's theorem and $\sigma$-finiteness?

I'm reviewing my analysis notes, and I am really confused about what is meant by $\sigma$-finiteness being a hidden hypothesis of Fubini's theorem. Here is Fubini's theorem as was stated to me: ...
4
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1answer
93 views

Replacing order of infinite sum and limit with some weaker form of dominated convergence

Let $f(x,n)$ be some positive function of $\mathbb R\times\mathbb N$ Using dominated convergence theorem, with $\mu(A)=|A|$ as the measure, we can prove that if $f(x,n)<g(n)$ for any $x$, and ...
2
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0answers
33 views

Existence of a nonmeasurable set with slices of measure zero

Does there exist a nonmeasurable set $A\subseteq \mathbb{R}^2$ such that for each line $l$ in the plane the intersection $A\cap l$ has one dimensional measure zero. (What about the problem for only ...
6
votes
2answers
58 views

Existence of a random variable satisfying a condition on its distribution

Let $X, Y : [0,1] \to \mathcal{X}$ be two random variables. Here, $[0,1]$ is the interval with the Lebesgue $\sigma$-algebra and $\mathcal{X}$ is a topological space with the Borel $\sigma$-algebra. ...
2
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1answer
43 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 ...
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1answer
19 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 ...
17
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1answer
417 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
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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
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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 ...
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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 ...
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1answer
29 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 $B_i$ ...
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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 $$F_X(x)=X_\ast P(]-\infty,x])=P(X\in]-\infty,x])$$ ...
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0answers
32 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
55 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 ...
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0answers
32 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
32 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
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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 ...