Articles of measure theory

An algebra (of sets) is a sigma algebra iff it is a monotone class

Let $X$ be a set and $\mathcal{A}$ an algebra of sets of $X$. Show that $\mathcal{A}$ is a $\sigma$-algebra iff it is a monotone class. The fact that a $\sigma$-algebra is a monotone class is trivial; $\sigma$-algebras are closed under countable unions and intersections, so it is in particular closed under monotone limits. It is […]

Understanding the measurability of conditional expectations

My question is about the conditional expectation of random variables with respect to a $\sigma$-algebra. I am having trouble getting an intuition behind the definitions among other things. I know that if $X$ is $\mathcal{G}$-measurable then $\mathbb{E} [X| \mathcal{G}] = X$, but what if $X$ is not $\mathcal{G}$-measurable? Is this expression just not defined? Furthermore, […]

the measurability of $\int_0^t X(s)ds$

For a $F_t$ adapted process $X$, please prove that $\int_0^t X(s)ds$ is $F_t$ measurable. For simple processes $X$, the conclusion is obvious. I think we should use monotone class theorem to prove the case of general processes. However, I don’t know the details. Thank you.

Show that the Fourier Transform is differentiable

This question already has an answer here: Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$ 1 answer

Tight sequence of processes

Let $X_{n} \in \mathbb{R}^{\infty}$ be a tight sequence of processes in metric space $(\mathbb{R}^{\infty}, l_{2})$ and for each $x\in\mathbb{Z}_{+}$ we have that $X_{n,x}\stackrel{d}{\to} Y_{x}$. Does it follow the weak convergence of the process $X_{n}$ to $Y$?

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k \hookrightarrow \mathbb{R}^n$ we have $\mu_{k}(U \cap i(\mathbb{R^k}))=0$. Does this imply $\mu_n(U)=0$?

Borel Measures: Atomic Decomposition

Context The notion of atoms and point masses agree to certain extent. (See Summary on Atoms.) Measures decompose w.r.t. atoms. (See Paper on Atoms.) Here, the goal is a direct approach to decompose w.r.t. point masses! Problem Consider a sigma-finite measure $\mu:\Sigma\to\mathbb{R}_+$. Does it decompose into a discrete and a continuous part: $$\mu=\mu_0+\mu_\infty$$ (For a […]

Finitely but not countably additive set function

Let X be any countable! set and and let F be the cofinite set, i.e., $A \in F $ if A or $A^{c}$ is finite (this is an algebra). Then show that the set function $\mu: F \rightarrow [0,\infty)$ defined as $\mu(A)=0$ if A is finite $\mu(A)=1$ if $A^{c}$ is finite is finitely additive. I […]

If $\mathbb E=0$ for all $G\in \mathcal G$, does $X=0$?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $\mathcal G$ a subfield of $\mathcal F$. I have that $\mathbb E[X\boldsymbol 1_G] = 0$ for all $G\in \mathcal G$. Do we have that $X=0$ ? I proved that if $X\geq 0$ a.s. then $X=0$ a.s. but if $X$ is just measurable, then I have that […]

Dynkin's Theorem, and probability measure approximations

I am working my way through Resnick’s A Probability Path, and looking at Exercise 2.5 I am a bit stuck on the application of Dynkin here. The question states: Problem: Let P be a probability measure on $ \mathcal{B}(\mathbb{R})$ and. For any $B\in \mathcal{B}(\mathbb{R}),\varepsilon>0 $ there exists a finite union of intervals, A, such that […]