# Tagged Questions

Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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### Kolmogorov 0-1 Law Converse?

Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Conjecture: Suppose we have events $A_1, A_2, ...$ s.t. $\forall \ A \in \bigcap_n \sigma(A_n, A_{n+1}, ...)$, $P(A) = 0$ or $1$. ...
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### Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that A_{S_k \...
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### Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)?

Why is ${|f_n-f|^p}$ uniformly integrable and tight iff {$|f_n|^p$} is uniformly integrable and tight ($f_n \rightarrow f$ pointwise)? This is from the last sentence in the proof in the following ...
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### Null Laplace Transform

As the title says, if I had a real signed measure $\nu$ defined on Borel sets of $\mathbb{R}^m$ with Laplace Transform vanishing on every $m$-tuple, can I say that $\nu=0$?
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### Real Analysis, Folland Proposition 2.21 Integration of Complex Functions

Proposition 2.21 - The set of integrable real-valued functions on $X$ is a real vector space, and the integral is a linear functional on it. Attempted proof - Note that we can derive the axioms of a ...
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### Real Analysis, Folland The Dominated Convergence Theorem

Background Information: Proposition 2.16 - If $f\in L^+$, then $\int f = 0$ iff $f = 0$ a.e. Question: 2.24 The Dominated Convergence Theorem - Let $\{f_n\}$ be a sequence in $L^1$ such that ...
Background Theorem 2.36 of Folland's Real Analysis says that if $(X,M,\mu)$ and $(Y,N,\nu)$ are sigma finite measure spaces, and $E\in M\bigotimes N$, then $x\mapsto \nu(E_x)$ and $y\mapsto \mu(E^y)$ ...
If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to \$\mu\...