Let $ \left\{ {r_i } \right\}_{i = 1}^\infty = \mathbb{Q}$ an enumeration of $\mathbb{Q}$. Let $ J_{n,i} = \left( {r_i – \frac{1} {{2^{n + i} }},r_i + \frac{1} {{2^{n + i} }}} \right) $ $ \forall \left( {n,i} \right) \in {\Bbb N}^2 $ Then define $$ A_n = \bigcup\limits_{i \in {\Bbb N}} {J_{n,i} } $$ […]

$\bf{\text{(Pettis Measurability Theorem)}}$ Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure. The following are equivalent for $f:\Omega\to X$. (i) $f$ is $\mu$-measurable. (ii) $f$ is weakly $\mu$-measurable and $\mu$-essentially separately valued. (iii) $f$ is Borel measurable and $\mu$-essentially separately valued. $\bf{\text{Relevant Definitions:}}$ A function $f:\Omega\to X$ is simple if it assumes only finitely many values. That is, […]

The maximal sigma algebra on a set is its power set. When the set is countable, its maximal sigma algebra can be generated by all singleton subsets, i.e. subsets each consisting of exactly one element. Conversely, if the maximal sigma algebra on a set can be generated by all singleton subsets, must the set be […]

Suppose that $E_1,E_2$ are two measurable (Lebesgue) subsets of $R^d$. Define $E=E_1\times E_2=\left\{(x,y)|x\in E_1, y\in E_2\right\}$. Can we say that $E$ is a Lebesgue measurable subset of $R^{2d}$? Recall (cf. Stein’s Real analysis) that a subset $E$ of $R^d$ is called measurable if for any $\epsilon>0$, there exist an open set $O\supset E$, such that […]

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different proof that only works when $X$ is sigma finite, but then it establishes also that the dual of $L^1(X)$ is […]

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on $S$. I’d appreciate some quidance on proving that $\mathcal{B}(S)$ is generated by the collection of sets of the form $$ \{f: (f(t_1),\ldots, f(t_n)\in B_1\times\cdots \times B_n\}, $$ where $B_j\in \mathbb{B}(\mathcal(R))$, $n<\infty$ […]

Consider the following random variables $$X:\Omega\to\mathbb{R}\quad\text{and}\quad Y:\Omega\to \mathbb{R}$$ and $$Z:=XY$$. One may interpret it as follows, i.e. $$Z(\omega) = X(\omega)Y(\omega).$$ In general, we cannot say much about the relations amongst $\sigma(X),\sigma(Y)$ and $\sigma(Z)$, as discussed in the answer below. However, if we consider $X_i:\Omega \to \mathbb{R}$ for $i\in(\infty,\infty)$. Why do we have $\sigma(\cdots,X_{n-1}X_n)\subsetneq \sigma(\cdots,X_n)$?

Let $S_1$ and $S_2$ be sets. Let $n_1$ be the cardinality of $S_1$ and $n_2$ be the cardinality of $S_2$. I assume that $n_1$ and $n_2$ are finite. Let $e$ be a function that maps members of $S_1$ and $S_2$ to real numbers. Assume that we have: $(1/n_1) \sum\limits_{x_i \in S_1} e(x_i) \geq (1/n_2) \sum\limits_{x_i […]

In his nice answer about uniform integrability, Did did show that condition (C) below is equivalent to (C1) and (C2) together. (C) For every $\varepsilon\gt0$, there exists a finite $c$ such that, for every $X$ in $\mathcal H$, $\mathrm E(|X|:|X|\geqslant c)\leqslant\varepsilon$. (C1) There exists a finite $C$ such that, for every $X$ in $\mathcal H$, […]

Trying to solve exercise 1.1.18 in D.W. Stroock, Probability Theory, I somehow don’t see how to get the hint in that exercise. Given a set $\Omega$, a tail $\sigma$-algebra $\tau$ generated by $\sigma$-algebras $\cal F_n$, where each $\cal F_n$ is again generated by a set $A_n$, that is ${\cal F}_n = \{\emptyset, \Omega, A_n, \Omega […]

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