The Annals of Probability

Conditional Distributions as Derivatives

P. Pfanzagl

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx}
Secondary: 28A15: Abstract differentiation theory, differentiation of set functions [See also 26A24]

Keywords
Conditional distributions differentiation of measures

Citation

Pfanzagl, P. Conditional Distributions as Derivatives. Ann. Probab. 7 (1979), no. 6, 1046--1050. doi:10.1214/aop/1176994897. https://projecteuclid.org/euclid.aop/1176994897


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