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Homework 5, Real Analysis 2

Due March 9


Problem 1


Let f be integrable, and for each α>0 let Eα={x:|f(x)|>α}. Then

|f|=0m(Eα)dα.



Problem 2 (Riemann-Lebesgue Lemma)


For fL1(\Rd), let

ˆf(ξ)=\Rdf(x)e2πixξdx


be its Fourier transform. Then ˆf(ξ)0 as |ξ|0.

Problem 3


Let f,gL1(\Rd).

  1. (x,y)f(xy)g(y)L1(\Rd×\Rd).

  2. The convolution fg(x)=\Rdf(xy)g(y)dy is well defined for a.e. x.

  3. ||fg||L1||f||L1||g||L1.

  4. ^fg(ξ)=ˆf(ξ)ˆg(ξ).


Problem 4


There does not exist IL1(\Rd) such that, for all fL1(\Rd),

fI=f.



Problem 5


Let fnf in measure.

  1. If fn0, then flim inf

  2. If there exists g\in L^1 such that |f_n|\le g for all n, then f_n\to f in L^1 and \displaystyle \int f_n \to \int f.

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