A regular conditional measure ν
on a space Y relative to an outer measure μ on a space X is defined as a function on
X ×ℛ such that (1) for each x ∈ X,ν(x,⋅) is an outer measure on Y and ℛ is the
family of subsets of Y which are (Carathéodory) measurable under each of the
measures ν(x,⋅),x ∈ X, and (2) for each β ∈ℛ the function ν(⋅,β) on X is μ
integrable) i.e., ∫
ν(x,β)μdx ≦∞.
Letting g be the function on the subsets of Z = X × Y defined by
defining a covering family F to consist of those rectangles A × B where A is μ
measurable, B ∈ℛ and g(A×B) < ∞ or those sets N such that g(N) = 0, we obtain
the outer measure ϕ = (μ∘ν) on Z generated by (the content) g and covering family
F.
A system of regular conditional measures is a sequence begun by a measure ν_{0} on
a space X_{1} and followed by regular conditional measures ν_{i} (relative to μ_{i}) on spaces
X_{i+1}(i = 1,2,⋯) where μ_{1} = ν_{0} and μ_{i+1} = (μ_{i} ∘ν_{i}) for i = 1,2,⋯ . Set X = ∏
_{x}X_{i},
and for x ∈ X write x^{i} for the point (x_{1},x_{2},⋯,x_{i}) which is the projection of x onto
the space X^{i} = ∏
_{j=1}^{i}X_{j} and similarly write S^{i} = ∏
_{j=1}^{i}S_{j} whenever the sets S_{j} are
subsets of X_{j}(j = 1,⋯,i).
For such a system of regular conditional measures a generalization of Tulcea’s
extension theorem for regular conditional probabilities holds, a Fubinilike
theorem for integrable functions is obtained and finally, for topological spaces, a
condition is given for the extension of inner regularity and almost Lindelöfness
properties.
