Question

Proposition 4.1:

Here is the question:

Proposition 4.1. Let (E,E. u) be a measure space. Then the following holds for all measurable sets A, B and A, A2,.. (we do not require them to be disjoint): (Finite additivity): AnB = 0 (AUB) /(A)ja(B), (Monotonicity): A C B u(A) < i(B) (A) (Sequential continuity): An A >u(An) If u(A1) and A,\A, then u(An)/(A) (U An) < E1 4(An). (Boole's inequality): fa

Answer 1

Here we firstly read and understand all properties (statement) in above theorem

Then read carefully

we know that (E,E) bo ole no + Hero manre pace then felleing ae hold for al) megou1 able 의 AIp f A1s Ar LHere we do not tequred them to be dioo6). d that s the Gequenlia) Continuity An AA JLAN) AALA) 2Zump Ta de YA,then uCAn) VKA) know AT) Coithout uA) D , the above result 16 Catake 2nd concluoion withoul asoumption An fatl, ie R JR I An= (ni) C IR for al n nil thal m(Ai) In example but Here oo ta ke at n AI-(ni) c IR then An yA An) y LA)