I`m having some trouble with how to prove that if μ is finite then it is sum-finite.I know that if μ is finite on a measurable space (E,A) if μ &lt; ∞ and that μ is sum-finite on a measurable space (E,A) if μ = ∑ n = 1 ∞ μ n , where μ n , for n ∈ N , is finite measures. But I dont know how to tie these definitions together.

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I`m having some trouble with how to prove that if   μ is finite then it is sum-finite.I know that if   μ is finite on a measurable space (E,A) if   μ  &amp;lt;  ∞ and that   μ is sum-finite on a measurable space (E,A) if   μ  =      ∑          n      =      1              ∞            μ    n  , where       μ    n  , for   n  ∈      N  , is finite measures. But I dont know how to tie these definitions together.

Dwayne James · Accepted Answer

Another solution, choose your favorite positive sequence   (      a    n        )          n      ∈              N             that sum to 1, and let       μ    n    =      a    n    μ simply be a scaled version of   μ. Then for all elements   B of the   σ-algebra   A we find:      ∑          n      =      1        ∞        μ    n    (  B  )  =      ∑          n      =      1        ∞        a    n    μ  (  B  )  =  μ  (  B  )      ∑          n      =      1        ∞        a    n    =  μ  (  B  )  .

Hector Petersen · Accepted Answer

The trivial measure   ζ that assigns 0 measure to all sets is finite.Solution:  μ  =  μ  +      ∑          n      =      2        ∞    ζis an expression for   μ as a countable sum of finite measures.

I`m having some trouble with how to prove that if μ is finite then it is sum-

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