Articles of measure theory

R as a union of a zero measure set and a meager set

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 $$ A_n = \bigcup\limits_{i \in {\Bbb N}} {J_{n,i} } $$ […]

Contradiction achieved with the Pettis Measurability Theorem?

$\bf{\text{(Pettis Measurability Theorem)}}$ 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, […]

When can the maximal sigma algebra be generated by all singleton subsets?

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 […]

Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?

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 […]

Duality of $L^p$ and $L^q$

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 […]

Generating the Borel $\sigma$-algebra on $C()$

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 $B_j\in \mathbb{B}(\mathcal(R))$, $n<\infty$ […]

inclusion of $\sigma$-algebra generated by random variables

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 $\sigma(\cdots,X_{n-1}X_n)\subsetneq \sigma(\cdots,X_n)$?

Affine transformation

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 […]

Why is this weaker then Uniform Integrability?

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 $\mathcal H$, […]

Atoms in a tail $\sigma$-algebra as $\liminf C_n$

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 […]