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

Result R3589 on D4381: Random basic number
Subresult of R2803: Basic Borel function almost everywhere finite if integral is finite

Random basic number almost surely finite if first absolute moment is finite

Formulation 0

Let $X \in \text{Random} [-\infty, \infty]$ be a D4381: Random basic number such that

(i)	\begin{equation} \mathbb{E} \|X\| < \infty \end{equation}

Then \begin{equation} |X| \overset{a.s.}{<} \infty \end{equation}

Formulation 1

Let $X \in \text{Random} [-\infty, \infty]$ be a D4381: Random basic number such that

(i)	\begin{equation} \mathbb{E} \|X\| < \infty \end{equation}

Then \begin{equation} \mathbb{P}(|X| < \infty) = 1 \end{equation}

Subresults

▶	R5008: Random unsigned basic number almost surely finite if expectation is finite

Proofs

Proof 0

Let $X \in \text{Random} [-\infty, \infty]$ be a D4381: Random basic number such that

(i)	\begin{equation} \mathbb{E} \|X\| < \infty \end{equation}