1).If A is a subset of B, A,B in m ( measurable sets) then show that integral (region A) s dM =< integral ( region B) s dM
Where s here is a simple non-negative measurable function. ( Please don't confuse this with bounded measurable functions, I need the proof for SIMPLE functions).
2). If E are measurable, X_E is the charachteristic function, s is as defined in 1, then show that
integral (region E) s d M = integral (region X) X_E s D M.
Measurable Sets and Spaces and Properties of Integrals of Simple Functions are investigated. The solution is detailed and well presented.