nLab almost surely

Definition

In probability theory an event of probability 1 is said to happen almost surely.

That is,

A statement $S$ about outcomes is said to be true almost surely, or with probability 1, if

$F := \{ \omega : S(\omega) \ \text{is true} \ \} \in \mathcal{F}$ and $\mathbf{P}(F) = 1$.

Where $\omega$ is a point in the sample space $\Omega$, $\mathcal{F}$ a $\sigma$-algebra? on $\Omega$, and $\mathbf{P}$ a probability measure on $(\Omega$, $\mathcal{F})$. (Williams 1991)

In measure theory, such subsets are also known as full subsets. Their complements are known as null subsets or negligible subsets.

Related concepts

• null subset

• full subset

• deterministic random variable

References

See also:

• Wikipedia, Almost surely

• Williams, D. (1991). Probability with Martingales. Cambridge: Cambridge University Press. ISBN:9780521406055