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Journal Article

# Conditional Distributions as Derivatives

P. Pfanzagl
The Annals of Probability
Vol. 7, No. 6 (Dec., 1979), pp. 1046-1050
Stable URL: http://www.jstor.org/stable/2243105
Page Count: 5

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Let $(X, \mathscr{a}, P)$ be a probability space, $Y$ a complete separable metric space, $Z$ a separable metric space, and $s: X\rightarrow Y, t: X\rightarrow Z$ Borel measurable functions. Then the weak limit of $P\{s \in B, t \in C\}/P\{t \in C\}$ for $C\downarrow\{z\}$ exists for $P-\mathrm{a.a.} z \in Z$, and is a regular conditional distribution of $s$, given $t$.