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I am currently working through a proof in my Real Analysis book, by Royden/Fitzpatrick and I'm confused on a part.

if f is a measurable function on E, f is integrable over E, and A is a measurable subset of E with measure less than δ, then ∫|f| < ε

A

Proof: for c>0

∫f = ∫f + ∫f <= (c)(m(A)) + 1/c ∫f

A {x in A s.t. f(x)< c} {x in A s.t. f(x)>=c} E

I understand why the first integral is less than (c)(m(A)) but I don't understand the second part.

Chebychev's inequality says that

if f is a non-negative measurable function on E then for any λ > 0

m{x in E s.t. f(x) >= λ} <= (1/λ)∫f

E

so here we would have that

m{x in A s.t. f(x)>=c} <= (1/c)∫f

A

and I don't understand how the book went from this step to getting that

∫f <= 1/c ∫f

{x in A s.t. f(x)>=c} E

any help is appreciated...thanks!

EDIT: for some reason the integrals aren't lining up with the sets they are being integrated over, hopefully it is still readable, if not please ask

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# Quick Chebychev Inequality Question

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