BV functions and HahnÃ¢Â€Â“Jordan decomposition

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# BV functions and HahnÃ¢Â€Â“Jordan decomposition

Hi,

I know that every BV function f on the reel can be written as the difference of two non decreasing functions g,h i.e f = g-h

The HahnÃ¢Â€Â“Jordan decomposition Theorem says that every signed measure v an be written as the difference of two positive measures m,n i.e v=m-n

I know that the the first statement follows from the second. Does this also imply that g and h are non negative?

I think it true because v(x)=f(x) dx is a signed measure and can therefore be written v(x)=m(x)-n(x). But since v is absolutely continuous so are m and n, therefore there are functions g,h such that m(x)=g(x)dx and n(x)= h(x)dx. This of course implies that f,g are non negative. But I unfortunately I do not know why f,g should be non decreasing?

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