The Annals of Probability

Conditional Distributions as Derivatives

P. Pfanzagl

Abstract

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 $t$.

Article information

Source
Ann. Probab., Volume 7, Number 6 (1979), 1046-1050.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994897

Digital Object Identifier
doi:10.1214/aop/1176994897

Mathematical Reviews number (MathSciNet)
MR548898

Zentralblatt MATH identifier
0427.60003

JSTOR