Articles of measure theory

Equivalent measures if integral of $C_b$ functions is equal

Is it true that if $X$ is a measure space and $\mu, \nu$ are Borel probability measures on $X$ if $$ \int_X \phi \ d \mu = \int_X \phi \ d \nu \qquad \forall \phi \in C_b(X) \text{ (continuous and bounded functions)} $$ then $$ \mu = \nu \text{ ?}$$ If $E$ is a measureable […]

Fourier Uniqueness Theorem: Proof?

I need this as lemma. Given the Borel space $\mathcal{B}(\mathbb{R})$. Consider a complex measure: $$\mu:\mathcal{B}(\mathbb{R})\to\mathbb{C}$$ Then one has: $$\int_{-\infty}^{+\infty}e^{it\lambda}\mathrm{d}\mu(\lambda)=0\implies\mu=0$$ How can I prove this?

convergence of step functions in $L^1$ norm

Let $f \in L^1 (m)$. For $k=1,2,3,…$, let $f_k$ be the step function defined by $$ f_k (x) = k\int_{j/k}^{\frac{j+1}{k}} f(t)dt \ \text{ for $\frac{j}{k}<x<\frac{j+1}{k}$, $j=0,\pm1,\cdots$.} $$ Show that $f_k$ converges to $f$ in $L_1$ norm. This one seems more direct but when I do the $\|f_n – f\|_1$, I have trouble switching the two […]

Dominated convergence for sequences with two parameters, i.e. of the form $f_{m,n}$

Let $f_{m,n}(x)$ be a sequence (dependent on $m$, $n$) of Lebesgue integrable functions on $\mathbb{R}$. Suppose that $f_{m,n}(x)\to 0$ as $m,n\to+\infty$, for almost $x\in\mathbb{R}$; in addition, $\left|f_{m,n}(x)\right|\le g(x)$ for all $m,n\in\mathbb{N}$, for all $x\in\mathbb{R}$, where $g\in L^1(\mathbb{R})$. Can we apply the Lebesgue dominated convergence theorem to conclude that $\int_\mathbb{R} {{f_{m,n}}\left( x \right)dx} \to 0$ as […]

(dis)prove:$\sup_{F \in 2^{(L^1(S,\mathbb{R}))}}\limsup\sup_{f\in F}|\int f dP_n-\int fdP|=\limsup\sup_{f\in L^1(S,\mathbb{R})}|\int fdP_n-\int fdP|$

Let $(S,d)$ be a complete separable metric space and consider the set $L^1(S,\mathbb{R})$ of functions $f:S \rightarrow \mathbb{R}$ which are 1-Lipschitz, i.e. $\forall x,y \in S: |f(x) – f(y)| \leq d(x,y)$. Further let: $$ \mathcal{P}^1(S) := \{P: \mathcal{B}_S \rightarrow [0,1] \mid P \mbox{ probability measure:} \forall a\in S: \int d(a,x) P(dx) < \infty \}, $$ […]

Proof that the preimage of generated $\sigma$-algebra is the same as the generated $\sigma$-algebra of preimage.

This question has also been asked here, but the answer there didn’t help me. I am trying to prove that, given some measurable space $(X, \Sigma)$, if $G$ is a collection of subset of $X$ such that $\sigma(G) = \Sigma$, then $$f^{-1}(\sigma(G)) = \sigma(f^{-1}(G)).$$ So far, I have been able to show that $\sigma(f^{-1}(G)) \subseteq […]

Show that $\mathfrak{S}=\bigcup_{N=1}^{\infty}\mathfrak{Z}_N\cup\left\{\emptyset\right\}$ is a semi-ring

Let $\Gamma$ be a finite set, $\Omega=\Gamma^{\mathbb{N}}=\left\{(x_1,x_2,\ldots):~\forall i\in\mathbb{N} x_i\in\Gamma\right\}$. For $a_1,\ldots,a_N\in\Gamma$ let $$ [a_1,\ldots,a_N]:=\left\{(x_1,x_2,\ldots)\in\Gamma^{\mathbb{N}}: i=1,\ldots,N x_i=a_i\right\} $$ be the $N$-cylinder which is determined by $a_1,\ldots,a_N$. Define $$ \mathfrak{Z}_N:=\left\{[a_1,\ldots,a_N]: a_1,\ldots,a_N\in\Gamma\right\}. $$ Show, that then $$ \mathfrak{S}:=\bigcup_{N=1}^{\infty}\mathfrak{Z}_N\cup\left\{\emptyset\right\} $$ is a semi-ring for $\Omega$. Hello! Three things are to show: (1) $\emptyset\in\mathfrak{S}$ (2) $A,B\in\mathfrak{S}\implies A\cap B\in\mathfrak{S}$ (3) $A,B\in\mathfrak{S}$ […]

Understanding Fatou's lemma

I want to prove that (without using Fatou’s lemma) for every $k \in N$ let $f_k$ be a nonnegative sequence $f_k(1),f_k(2),\ldots$ $$\sum^\infty_{n=1}\liminf_{k \to \infty} f_k(n) \le \liminf_{k \to \infty} \sum^\infty_{n=1}f_k(n)$$ Can you give some hint for me about that? hat

Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ $\forall i=\{1,\dots,N\}$. Then consider the product set $Y = X_1\times\dots\times X_i\times\dots\times X_N$ and say we are interested in a particular distribution $f_Y$, which is in […]

separation theorem for probability measures

Suppose I have a probability measure $\nu$ and a set of probability measures $S$ (all defined on the same $\sigma$-algebra). Are the following two statements equivalent? (1) $\nu$ is not a mixture of the elements of $S$. (2) There is a random variable $X$ such that the expectation of $X$ under $\nu$ is less than […]