Saturday, 20 August 2016

real analysis - Finite positive measure




I currently don't find any approach to this question. In particular, how can I related a finite positive measure to a regular integration?



Suppose μ be a finite positive measure. Let f be measurable, and g is an increasing function in C1. I am trying to prove the following property: g(f)dμ=0μ{x: |f(x)|>t}g(t)dt.


Answer



Just use Fubini:
0μ{x:|f(x)|>t}g(t)dt=01{|f|>t}(x)g(t)dμ(x)dt=01{|f|>t}(x)g(t)dtdμ(x)=|f(x)|0g(t)dtdμ(x)=g(|f(x)|)g(0)dμ(x)



So if g(0)=0 and you were searching for the property with g(|f|) instead of g(f) this is ok.


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