(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f be Lebesgue integrable on [0,a], and define g(x) = ∫(1/t)f(t)dt , lower integrand limit = x, upper integrand limit = a. Prove that g is integrable on [0,a], and that ∫f(x)dx = ∫g(x)dx .

2. Relevant equations

In the previous problem, I showed that if if f and g are measurable functions on σ-finite measure spaces (X,A,μ), (Y,B,[itex]\nu[/itex]), and h(x,y) = f(x)g(y), then h is measurable. Also, if f ε L^{1}(μ) and g ε L^{1}([itex]\nu[/itex]), then h ε L^{1}(μx[itex]\nu[/itex]) and ∫hd(μx[itex]\nu[/itex]) = ∫fdμ ∫gd[itex]\nu[/itex].

I think that I'm supposed to use these facts to solve this problem, but after multiple dead ends I'm really not sure how.

Oh, and I'm sure that Fubini's Theorem (∫hd(μx[itex]\nu[/itex]) = ∫[∫hd[itex]\nu[/itex]]dμ = ∫[∫hdμ]d[itex]\nu[/itex] will be used as well.

3. The attempt at a solution

Any help would be much appreciated : )

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# Product Measures, Fubini's Theorem

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