anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
@oldrin.bataku could you help me with this question please?
anonymous
  • anonymous
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
  • anonymous
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)

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anonymous
  • anonymous
is that correct for part a?
anonymous
  • anonymous
you should also show that scaling by \(c>0\) preserves nonnegativity but yes

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