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=∫∞0∫1{|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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