For a finitely additive and
countably multiplicative family H, Measurable H is the family of all sets which are
measurable by every Carathéodory outer measure by which the members of H are
measurable and complements of members of H are approximable from within. A
relation contained in a topological product space is subvalent, if for some countable
ordinal α, each horizontal section of the relation has an empty derived set of
order α. A topological space is Borelcompact if it and the difference of any
two of its closed compact subsets are countable unions of closed compact
sets.
It is shown that if X and Y are Borelcompact, Hausdorff spaces with countable
bases and R is an analytic and subvalent subset of the cartesian product of X with
Y , then the direct R-image of A is Measurable F(Y ) whenever A is Measurable
F(X). (F(X) is the family of closed subsets of X.) If X and Y are complete,
separable, metric spaces and R is an analytic and subvalent subset of X × Y , the
same conclusion can be drawn.
