An alternate development of conditioning

Abstract The “classical” development of conditioning, due to Kolmogorov, does not agree with the “practical” (more intuitive, but unrigorous) way in which probabilists and statisticians actually think about conditioning. This paper describes an alternative to the classical development. It is shown that standard concepts and results can be developed, rigorously, along lines, which correspond to the “practical” approach, and so as to include the classical material as a special case. More specifically, let Xand Y be random variables (r.v.‘s) from (Ω, f, P) to (x, f_x) and (y. f^y.), respectively. In this paper, the fundamental concept is the conditional probability P(AX = x), a function of xε x which satisfies a “natural” defining condition. This is used to define a conditional distribution P_y/x, as a mapping x × f^y-R such that, as a function of B, P_ylx=x,(B) is a probability measure on f^y. Then, for a numerical r.v. Y, conditional expectation E(Y/X) is defined as a mapping x →r̄ whose value at x isE(Y/X = x) = ydP_Y/x=i(y). Basic properties of conditional probabilities, distributions, and expectations, are derived and their existence and uniqueness are discussed. Finally, for a sub-o-algebra and a numerical r.v. Y, the classical conditional expectation E(Y) is obtained as E(Y/X) with X = i, the identity mapping from (Ω, f) to (Ω).