21. See non-measurable set for more details. 22. Whenever " A " and " B " are any measurable sets and ? is the associated measure. 23. The ?-measurable sets form a ?-algebra and ? restricted to the measurable sets is a countably additive complete measure. 24. The ?-measurable sets form a ?-algebra and ? restricted to the measurable sets is a countably additive complete measure. 25. Halmos ) as a measurable set of positive measure that has no subsets of strictly less, yet positive measure. 26. Show that the set of points for which \ { f _ n \ } converges is a measurable set . 27. Specifically, this is true if the measure space contains an infinite family of disjoint measurable sets of finite positive measure. 28. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. 29. In ZF, one can show that the Hahn Banach theorem is enough to derive the existence of a non-Lebesgue measurable set . 30. A measure is called " ?-finite " if can be decomposed into a countable union of measurable sets of finite measure.
