I need help please to prove some properties of a positive measure which are (a)Let µ be a positive measure on a measurable space (X,M). Let c 0. Prove that ˜µ = cµ is also a measure (b) Let µ1, µ2 be two positive measures on a measurable space. Assume also that there is at least one set A ∈ M such that µ1(A) + µ2(A)

Question

I need help please to prove some properties of a positive measure which are

(a)Let µ be a positive measure on a measurable space (X,M). Let c > 0. Prove that ˜µ = cµ is also a measure

(b) Let µ1, µ2 be two positive measures on a measurable space. Assume also that there is at least one set A ∈ M such that µ1(A) + µ2(A) < ∞.

Prove that ˆµ = µ1 + µ2 is also a measure

anonymous · Answer

@oldrin.bataku could you help me with this question please?

anonymous · Answer

this is a good place to start http://www.math.uah.edu/stat/prob/Measure.html https://en.wikipedia.org/wiki/Measure_%28mathematics%29 but i still can't solve it!

anonymous · Answer

Given µ is a positive measure.
Define cµ = c* (µ (A) ) , where A is a set .
Claim: cµ is a positive measure
Proof:
1. cµ ( ∅ ) =  c * ( µ ( ∅ )  ) = c * 0 = 0

2. cμ(⋃i∈IAi)=∑i∈Icμ(Ai)
cμ(⋃i∈IAi)=c⋅(μ(⋃i∈IAi))=c⋅∑i∈Iμ(Ai)=∑i∈Icμ(Ai)

anonymous · Answer

is that correct for part a?

anonymous · Answer

you should also show that scaling by $c>0$ preserves nonnegativity but yes