ThmDex – An index of mathematical definitions, results, and conjectures.

Result R3587 on D4381: Random basic number
Subresult of R3586: Basic function is finite almost everywhere iff positive and negative parts are

Random basic number almost surely finite iff positive and negative parts are

Formulation 0

Let $X \in \mathsf{Random}(\to [-\infty, \infty])$ be a D4381: Random basic number.

Then $|X| \overset{a.s.}{<} \infty$ if and only if

(1)	\begin{equation} X^+ \overset{a.s.}{<} \infty \end{equation}
(2)	\begin{equation} X^- \overset{a.s.}{<} \infty \end{equation}

Formulation 1

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to [-\infty, \infty]$ is a D4381: Random basic number on $P$

Then $\mathbb{P}(|X| < \infty) = 1$ if and only if

(1)	\begin{equation} \mathbb{P}(X^+ < \infty) = 1 \end{equation}
(2)	\begin{equation} \mathbb{P}(X^- < \infty) = 1 \end{equation}

Proofs

Proof 0

Let $X \in \mathsf{Random}(\to [-\infty, \infty])$ be a D4381: Random basic number.