Two functions $f(x)$ and $g(x)$ are called equi-measurable if $m(\{x:f(x)>t\})=m(\{x:g(x)>t\})$. Nondecreasing rearrangement of a function $f(x)$ is defined as $$f^*(\tau)=\inf\{t>0:m(\{x:f(x)>t\}\leq\tau\}.$$ Prove that $f^*(\tau)$ and $f(x)$ are equimeasurable.
nondecreasing rearrangement is equimeasurable
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real-analysis
measure-theory
decreasing-rearrangements
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10What are your thoughts on the matter? Do you see a point where to start? – 2011-12-30
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0This proof can be found in "Classical Fourier Analysis", Grafakos, Proposition 1.4.5 (12) – 2016-02-25