Counter-example to continuity of measure in uncountable unions

Abstract

Continuity of measure asserts that the measure of the union of an increasing sequence of sets is equal to the supremum of the measures of those sets. We provide counter examples in the case of uncountable unions. We construct the first counter example on the ordinal numbers, and we show that counterexamples also exist in the reals if we assume the continuum hypothesis.

7 pages