Vol. 18, No. 1, 1966

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Images of measurable sets

David Wilson Bressler and A. P. Morse

Vol. 18 (1966), No. 1, 37–55

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.

Mathematical Subject Classification
Primary: 28.10
Received: 25 July 1964
Published: 1 July 1966
David Wilson Bressler
A. P. Morse