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Questions tagged [measure-theory]

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

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Proving an increasing of a continuous function

Let $f$ be continuous on $[a,b]$, and assume that $D^-f\geq 0.$ Show that $f$ increasing on $[a,b]$.(Hint: first show that for a function $g$ on $(a,b)$, then apply this to the function $g=f+x\epsilon....
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2answers
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If $f_k \to f$ uniformly, then $J(f_k) \to J(f)$, where $J: \mathcal{C}_{\text{c}}(\mathbb{R}^d) \to \mathbb{R}$ ist linear and monotonous

Let $\mathcal{C}_{\text{c}}(\mathbb{R}^d) := \{f: \mathbb{R}^d \xrightarrow{\mathcal{C}} \mathbb{R}: \text{supp}(f) \text{ is compact.} \}$. I have a few questions regarding the proof following ...
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Proof that equality of measures on the $\pi-$system implies equality on $\sigma$ - algebra generated by it

Let $\nu$ and $\nu'$ be $\sigma$-finite measures on {$E,\varepsilon$}. $K\subset \varepsilon$ is $\pi-$system, such that $\exists\, E_i\in K, E=\cup_{i=1}^{\infty}E_i $ $\nu(E_i)<\infty$. Then if $...
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1answer
37 views

If $E$ is a Lebesgue measurable set in $\mathbb{R}$ with positive measure, the set $E-E = \{ x-y : x,y \in E \}$ is also measurable?

I know that the set contains an interval (the Steinhaus theorem), but I can't use this for answer the question, maybe someone have a hint to prove that's true or a counterexample for show that's false....
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Total Variation show the set of points that appear infinity many times has measure zero

Let $f$ be continuous function on an interval $\left[ a,b\right]$. Show that the total variation $V\left( f; \left[ a,b \right]\right)$ is given by $$V\left( f; \left[ a,b \right]\right) = \int_{-\...
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1answer
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Equality of measures on $C([0,1])$, if integrals over functions continuous w.r.t. pointwise convergence coincide

The problem I pose below is a simplified version of my actual problem: Instead of $C([0,1], \mathbb{R})$ I actually have $C([0,+\infty], H)$, where $H$ is some separable, (infinite-dimensional), real ...
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1answer
16 views

$X_1,X_2$ independent gamma-distributed random variables. Density of $Y:=\frac{X_1}{X_1+X_2}$

Let $X_1,X_2$ be two independent gamma-distributed random variables: $X_1 \sim \Gamma(a_1,b), X_2 \sim \Gamma(a_2,b)$. How can I determine the density of $$Y:=\frac{X_1}{X_1+X_2}$$ I don't really ...
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1answer
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Collection of $\mu^*$ measurable subsets contains all the null sets

In Richard Bass' Real Analysis, I am struggling to understand part of the proof of Theorem 4.6 on page 28. Theorem. If $\mu^∗$ is an outer measure on $X$, then the collection $\mathcal{A}$ of $\mu^...
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1answer
23 views

Proving a property of $L^\infty$ spaces

Let $f$ be continuous and bounded on $\mathbb{R}^d$. Show that $\|f\|_\infty=$ sup$\{|f(\vec{x})|: \vec{x}\in\mathbb{R}^d\}$ relative to Lebesgue measure. Basically I need to prove that $\|f\|_\infty$...
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1answer
25 views

If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?

Let $(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces $(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$ $(N_t)_{t\ge0}$ be a $\mathbb ...
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1answer
28 views

Egorov’s Theorem (?)

Let $(X, \mathbb A, m)$ be a measurable space and let $\{f_n : X \to \mathbb R\}_{n \in \mathbb N}$ be a sequence of Borel measurable functions. If such sequence converges $m$-almost everywhere to ...
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Question regarding motivation of spectral theorem for unitary operators

$\newcommand{\mc}{\mathcal}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ Theorem B.4 in Einsiedler and Ward's [EW] Ergodic Theory with a view towards Number Theory states the following: Theorem 1. ...
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Is it possible to prove Fubini’s Theorem without Dynkin’s Theorem or the Monotone Class Theorem?

Fubini’s Theorem for Lebesgue integrals states that if $X$ and $Y$ are Sigma-finite measure spaces then the integral of a (well-behaved) function $f(x,y)$ with respect to the product measure on $X\...
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1answer
19 views

Borel $\sigma$-algebra on $R^k$

I am trying to solve the following question: Let $(\mathbb{R}^k, \tau)$ be a topological space. Consider the classes of sets $\mathcal{O}_1 = \left\lbrace (a_1, b_1) \times \dots \times (a_k, b_k) \ ...
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1answer
25 views

Question about Vitali Covering

Show that the Vitali Covering Lemma does extend to the case in which the covering collection consists of nondegenerate general intervals. I do not understand what does "nondegenerate general ...
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1answer
43 views

Approximating a Set as Lebesgue Measurable with open and closed sets

Let $A \subseteq \mathbb R^{d}$, and $\bar{B^{d}}:=\{B \cup \bar{N} \subseteq \mathbb R^{d}: B \in B^{d},\exists N \in B^{d}, \lambda^{d}(N) = 0, \bar{N} \subseteq N\}$. Show that: $\forall \epsilon ...
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2answers
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Prove that $n^{-2}\sum_{k=1}^nS_k$ converges in probability

Let $\{X_n\}$ be a sequence of uncorrelated random variables with common mean $\mu$, such that $\sup_n$Var$(X_n)<\infty$. If $S_n=\sum_{k=1}^n X_k$, show that $n^{-2}\sum_{k=1}^nS_k$ converges in ...
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Predictable Proceses (Da Prato & Zabcyk)

I'm reading Da Prato & Zabcyzk (2014), and in their Proposition 3.7, they assert that if $\Phi$ is an adapted and stochastically continuous process taking values, I believe, in $L(U,H)$ ($U$ and $...
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How do we get the Borel sets / sigma-fields?

I am doing mathematical finance, and the authors always try to motivate the mathematical foundations of probability. Often they roughly explain measure, and then talk about Lebesgue measure over, ...
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Distribution of cartesian product of two random variables

Suppose there are two random variables, $X$ and $Y$. Each of the random variables subjects to its distribution respectively. Now, I have two approaches of sampling. Approach 1 Sample $X_1, ..., X_N$ ...
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1answer
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Lack of Understanding that a measure on a space and the same measure on that same space union an element of null space should be equal

Context: Let $X$ be a set and $\mu$ a measure on a $\sigma-$algebra $\dot{A} \subseteq \mathcal{P}(X)$ and $N$ the $\mu-$Null Set, such that $\forall n \in N,$ $\mu(n)=0$ Define $\bar{A}:= \{A \cup M ...
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Sums of Bernoulli random variables depending on a Poisson process is a Poisson process?

Let $X_t$ be a Poisson process with intensity $\lambda$ and $Z_i$~$Bern(p)$. All the R.Vs are independent. I want to prove that $$Y_t=Z_1+Z_2+...+Z_{X_t}$$ Is a Poisson process with intensity $\...
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1answer
33 views

Integrable if absolutely summable

Let $P(\mathbb N)$ the power set of $\mathbb N$ and $f$ the counting measure on $(\mathbb N, P(\mathbb N) )$. If $\{ a_n \}_{n \in \mathbb N}$ is a real-valued sequence, then, so the statement in my ...
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1answer
30 views

Sum of measure of sequences

Given $(X, \mathbb A, m)$ a measurable space and $\{E_i\}_{i \in \mathbb N} \subset \mathbb A$ a sequence such that $\sum_{i \in \mathbb N} m (E_i) < + \infty$. I want to show that $m$-almost ...
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2answers
23 views

Limits of product of function and measure

Let $(X, \mathbb A, m)$ be a measurable space and $f: X \to \mathbb R$ a Borel-measurable function. If $\psi: [0, + \infty) \to [0, + \infty)$ is monotone non decreasing and $\int \psi ( |f|) dm < +...
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Convergence in Probability is equivalent to “Cauchy in Probability”

Let $X_n,n\geq 1$ be a sequence of real random variables defined on a probability space. Show that the following are equivalent: (i)$X_n$ converges in probability to a real random variable $X$. (ii) ...
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1answer
26 views

Convergence in Distribution implies Convergence in Expectation uniformly for an equicontinuous family

If $X_n \xrightarrow{\text{d}} X$, then for any equicontinuous family $\{f_\theta:\theta\in \Theta\}$, satisfying sup$\{|f_\theta(x)|:x\in \mathbf{R},\theta\in \Theta\}\}<\infty$, prove that $E[f_\...
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prove that $E[f_θ(X_n)] → E[f_θ(X)]$ uniformly in $θ ∈Θ$

If $X_n → X$ (in distribution), then for any equicontinuous family $\{f_θ:θ ∈Θ\}$, satisfying $sup\{|f_θ(x)| : x ∈ R, θ ∈ Θ\} < ∞$ , prove that $E[f_θ(X_n)] → E[f_θ(X)]$ uniformly in $θ ∈Θ$ . My ...
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29 views

Convergence of indicator function

I've given a sequence of function from $\mathbb R \to \mathbb R$ and $f_n (x) = \frac{1}{n} \mathbb 1_{[n, + \infty]} (x) $, where $n \in \mathbb N$. Does $f_n$ converge? How to check the convergence ...
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1answer
21 views

Showing that a sequence of functions has no dominant (Dominated convergence theorem)

Define $u_n: \mathbb{R}^+ \to \mathbb{R}^+$ as $u_n(x) = n1_{(0,1/n)}$. I need to show that this has no dominant. A dominant is defined as an integrable function $w: X \to \mathbb{R}^+$ such that $\...
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0answers
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Integral over a set of measure 0

Let's $A$ be such that $\lambda (A) = 0$ (Lebesgue measure). I want to prove that for every measurable function $f$, $\int_A f(x) \lambda(dx) = 0$ I did the following : $|\int_A f(x) \lambda(dx)| &...
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0answers
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Do compacta in $\mathbb R^n$ have finite Hausdorff measure? [duplicate]

For $A\subset \mathbb R^n$ let $\mathcal H^s(A)$ be the $s$-dimensional Hausdorff measure with respect to the Euclidean metric. The Hausdorff dimension of $A$ is given by $$\dim_H(A) = \inf\{s>0\...
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1answer
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A question on the Borel sigma field on $C[0,1]$

Let $C$ denote the σ-field on $C[0,1]$ generated by all open balls : the open $\epsilon$-ball around $f$ is the set $\{g ∈ C : ρ(f, g) < \epsilon\}$ Denote $\forall t∈[0,1], π_t : C[0,1] →\Bbb ...
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1answer
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$f:[a,b] \to \mathbb{R}$ lipschitz continuous $\Leftrightarrow$ $\exists \ g:\|g\|_\infty<+\infty$ so that $f(x)=\int_{[a,x]}g \ d \lambda$

I want to show: $f:[a,b] \to \mathbb{R}$ lipschitz continuous $\Leftrightarrow$ $\exists \ g:\|g\|_\infty<+\infty$ so that $f(x)=\int_{[a,x]}g \ d \lambda$ Anyone got any hints how to prove ...
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Baire Category Theorem Example Problem difficulty

• Assign an integer, n, to each real number. Let A_n = {x ∈ R : x → n}, then R = U A_n, so one of the A_n’s must be somewhere dense. So each collection A_n would be a set of size Aleph-null? Then the ...
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1answer
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Is the projection of a Jordan measurable set Jordan measurable?

The projection of a Borel set need not be a Borel set, and the projection of a Lebesgue measurable set need not be Lebesgue measurable. My question is, is the projection of a Jordan measurable set ...
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1answer
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Lebesgue Measurable Set which is not a union of a Borel set and a subset of a null meager set?

This is a follow-up to my question here. The Lebesgue Sigma algebra is the completion of the Borel Sigma algebra under the Lebesgue measure, which means that every Lebesgue measurable set can be ...
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2answers
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Is every Borel subset of a measurable set measurable?

Let $A$ be a Lebesgue measurable subset of $\mathbb{R}$. Let us consider the subspace topology on $A$, and let us consider the Borel sigma algebra under that topology. My question is, is every Borel ...
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1answer
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A dense measure $0$ $G_\delta$ subset of the Fat Cantor set?

The fat Cantor set is a nowhere dense subset of $\mathbb{R}$ with positive Lebesgue measure. My question is, does there exist a $G_\delta$ set dense in the fat Cantor set with Lebesgue measure $0$? ...
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1answer
28 views

$f$ is increasing implies $Mf$ is increasing ???

For a given function $f:\mathbb{R} \rightarrow \mathbb{R} $, Hardy–Littlewood maximal function of $f$ defined by $$Mf(x)=\sup_{r>0}\frac{1}{2r}\int\chi_{[x-r,x+r]}|f|~d\lambda=\sup_{r>0}\frac{1}{...
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1answer
44 views

How can I show that the following function is measurable?

For $0\leq t_1<t_1<...<t_n \leq 1$, $n\in \mathbb{N}$ we define $$\pi_{t_1,...t_n}:(C[0,1],\mathbb{B}(C[0,1]))\to (\mathbb{R}^n,\mathbb{B}(\mathbb{R}^n)), \quad x\to (x(t_1),...,x(t_n)).$$ $\...
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2answers
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Measurable Operators

Suppose I have an operator valued function, $\omega\mapsto A(\omega)$; for each $\omega$, $A(\omega):X\to Y$, is a bounded linear operator with $X$ and $Y$ real Hilbert separable Hilbert spaces. ...
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1answer
62 views

A perfect nowhere dense set which intersects every open set with positive measure?

A perfect set is a closed set with no isolated points. A nowhere dense set is a set whose closure has empty interior. My question is, what is an example of a nonempty perfect nowhere dense subset of ...
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1answer
21 views

Question on sigma-Algebra of a Probability space

I am trying to understand what a sigma algebra is and I found this really helpful pdf online. On the last couple of lines on page 1 though, the author writes: "Any event E is a subset of Ω. Assume ...
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1answer
27 views

How to show that the Wiener measure is singular with respect to a normal law

We have a Gaussian process $X$, $X_t:=B_t - tB_1$, where $B$ is a $BM$, $t\in[0,1]$. Let $\nu$ be the law of $X$ and $\mu$ the Wiener measure. How can I show that $\mu$ is singular with respect to $...
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0answers
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Measure theory equalities [duplicate]

Let $(X,\mathbb{B}, \mu)$ be a measure space and let $(E_k)_{k=1}^\infty$ a sequence of measurable sets such that $\sum_{k=1}^\infty \mu(E_k)<\infty$. Now, for each $n\in\mathbb{N}$ we define $A_n$ ...
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0answers
26 views

Conditional expectation of a function with independent random variables

If $X_1, ..., X_{n+1}$ are independent real random variables and $h:\mathbb{R}^{n+1} \to \mathbb{R}$ a Borel function. Now taking the conditional expectation $\mathbb{E}[h(X_1, \ldots, X_{n+1})| \...
3
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2answers
28 views

Let $f: \Omega \to [0, \infty]$ be measurable. Let $\mu$ be $\sigma$-finite. Then $\{t\mid \mu\{f = t\} \neq 0 \}$ is countable.

Let $f: \Omega \to [0, \infty]$ be measurable. Let $\mu$ be $\sigma$-finite. Then $\{t\mid \mu\{f = t\} \neq 0 \}$ is countable. My attempt: I managed to show the result if $\mu(\Omega) < \infty$....
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1answer
26 views
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Verification of a semialgebra

The question concerns the verification that a set family is a semialgebra. Definition. A non empty family $\mathcal{S}\subseteq\mathcal{P}(X)$ is said semialgebra on $X$ if: $1.$ For each $E, ...
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1answer
36 views

Show that Riemann integral and Lebesgue integral coincide.

I'm proving that for $f: (\Omega, \mathcal{F}) \to [0, \infty]$ and a $\sigma-$finite measure $\mu$ on the $\sigma$-algebra $\mathcal{F}$, we have $$\int_\Omega f d\mu = \int_{0}^\infty \mu \{f \geq ...