Here (Section “Integration in respect to a complex measure”), they say that: Another approach is to not develop a theory of integration from scratch, but rather use the already available concept of integral of a real-valued function with respect to a non-negative measure. To that end, it is a quick check that the real and […]

Based on Williams’ Probability w/ Martingales: Let $(S, \Sigma, \mu)$ be a measure space. Dominated Convergence Theorem: Suppose $\{f_n\}_{n \in \mathbb{N}}$, $f$ are $\Sigma$-measurable $\forall n \in \mathbb{N}$ s.t. $\lim_{n \to \infty} f_n(s) = f(s) \forall s \in S$ or a.e. in S and $\exists g \in \mathscr{L}^1 (S, \Sigma, \mu)$ s.t. $|f_n(s)| \le g(s) […]

I am having trouble understanding the measurability issues arising with almost sure / almost everywhere convergence. $X_n \rightarrow X$ a.s. if $\Pr \{ \lim X_n = X \} = 1$. Put differently, $\forall \varepsilon > 0$ there must be an $N > 0$ such that $\forall n > N$, $\Pr \{ \omega : | X_n […]

Let $\mu$ be a fixed finite measure on $\mathbb R$. We say that $\mu$ is doubling if there exists a constant $C>0$, such that for any two adjacent intervals $I=[x−h,x]$ and $J=[x,x+h]$, $$C^{−1}\mu(I)≤\mu(J)≤C\mu(I).$$ Assuming that $\mu$ is doubling, show that there exist positive constants $B$ and $a$, such that for every interval $I$, $$\mu(I)≤B[length(I)]^a$$ By […]

Let $X,Y$ and $Z$ be Borel spaces (that is, Borel subsets of Polish spaces) and let $\mathcal P(X)$ denote the Borel space of all Borel probability measures on $X$. For a product measure $P\in \mathcal P(X\times Y\times Z)$ let $\kappa:X\times Y\to\mathcal P(Z)$ be a regular conditional probability on $Z$ given $X\times Y$. We say that […]

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to prove this, for each $n \in \mathbb{Z}$ let $E_n=E \cap (n,n)$. How to show that the regularity theorem for $E_n$ can […]

Can a countable union of non-measurable sets of reals be measurable? For instance, can we partition $\mathbb{C}$ into countably many disjoint non-measurable sets?

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, …, $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p’}(E)$, where $p’$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$ This is from a past qual. Not really sure […]

Let $Y$ be a collection of subsets of the set X. Show that for each $A \in \sigma(Y)$ there is a countable subfamily $B_0 \subset Y$ such that $A\in \sigma(B_0)$ My try: I look at $\cup B_i$ where $B_i$ is a countable subfamily of $Y$. And I want to show that $Y\subset \cup B_i \subset\sigma(Y)$. […]

Suppose that the measurable sets $A_1,A_2,…$ are “almost disjoint” in the sense that $\mu(A_i\cap A_j) = 0$ if $i\neq j$. Prove that $$\mu\left(\cup_{k=1}^\infty A_k\right)=\sum_{k\ge1}\mu(A_k)$$ Conversely, suppose that the measurable sets $A_1,A_2,…$ satisfy $$\mu\left(\cup_{k=1}^\infty A_k\right) = \sum_{k=1}^\infty\mu(A_k)<\infty$$ Prove that the sets are almost disjoint. Here $\mu(A)$ denotes the Lebesgue measure of $A$. I know that if […]

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