202405211636
Status: #idea
Tags: Probability Theory

Caratheodarry's Extension Theorem

Informally, it's a theorem that states that:
Say we have an algebra $Ξ$ defined on a sample space $Ω$ with some pseudo-measure $P_{0}$ (really just a function) that maps $Ξ$ to the interval $[0, 1]$ in such a way that:

$P_{0} (Ω) = 1$
$P_{0}$ is countably additive (if it so happen that the countable union of sets in $Ξ$ is in $Ξ$ )

Then it is uniquely extensible (as is there is exactly one actual probability measure) such that for that probability measure $P$ :

P_{0} (A) = P (A), for any A element of the σ - algebra under consideration

Note that at no point did we say that $Ξ$ had to be a $σ -$ algebra itself, hence the idea of extension.

Caratheodarry's Extension Theorem

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