We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

Keywords:

measure convex set, measure extremal set, face measure convex set, measure extremal set, face

MSC Classifications:

46A55, 52A07 show english descriptions Convex sets in topological linear spaces; Choquet theory [See also 52A07]
Convex sets in topological vector spaces [See also 46A55] 46A55 - Convex sets in topological linear spaces; Choquet theory [See also 52A07]
52A07 - Convex sets in topological vector spaces [See also 46A55]

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