CONDITIONAL PROBABILITY IN NONSTANDARD ANALYSIS

Vesna Musicki-Kovacevi\'c

Abstract: In this paper we apply the theory of Loeb measure to conditional probability for hyperfinite Loeb spaces. We show that conditional probability $^\sim P(\cdot/A)$ on a Loeb space $(V,\frak M(^\sim P), ^\sim P)$ for $A\in^\ast\frak B(V)$, $(P(A)> 0$ and $P(A)\not\approx 0$ is a Loeb measure and for $A\in \frak M(^\sim P)$ $(^\sim P(A)>0)$ can be represented by a Loeb measure. For the case $A\in \frak M(^\sim P)$ we prove that there exists a set $C\in ^\ast\frak B(V)$ such that $^\sim P(\cdot/A)$ is equal to the Loeb conditional probability $L(P(\cdot/C))$. We introduce internal conditional probability relative to an internal subalgebra $\frak U$ of $^\ast\frak B(V)$ as in case of finite standard probability spaces. We show, analogously to a well-known probability result, that internal conditional probability $P(A/\frak U)$, $A\in ^\ast \frak B(V)$, and internal conditional expectation $E(X/\frak U)$, $X$ is $S$-integrable, are $P$-a\. s\. unique, in nonstandard sense, random variables on $(V,\frak U,P)$. Finally, we give a nonstandard characterization of conditional probability $^\sim P(A/\frak M (\frak U))$, $A\in \frak M(^\sim P)$ on a Loeb space $(V,\frak M(^\sim P), ^\sim P)$. We prove that there exists a set $C\in ^\ast \frak B(V)$ such that $P(C/\frak U)$ is the lifting of $^\sim P(A/\frak M(\frak U))$.

Classification (MSC2000): 03H10

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