Show that if $\int fh < \infty$ for all $h \in L^q$ then $f \in L^p$

Asked May 05 '20 at 19:45

Active May 05 '20 at 20:09

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We can assume we are on Lebesgue measure if necessary. This seems like it should not be hard to show but this is the first problem i encountered which wants you to prove integrability. My idea was as follows. Let $A_n=\{x | |f(x)||>n\}$ then we have two options, either $\mu(A_n)<\infty$ for some $n$ or $\mu(A_n)=\infty$ for all $n$. The first case: By continuity of measure we know that for some big $M$ $\mu(A_M)<1$ and so it's characteristic function belong to $L^q$ and we know the integral over that set is finite. So now we have the function is bounded on an infinite set. I am not sure what to do now. My suspicion is pick a good $h$ but I do not know how. Hints would be appriciated.

asked May 05 '20 at 19:45

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Sign problems here. Note that "$<\infty$" does not mean "converges", except when the integrand is nonnegative. A better statement would be: $\int |fh| < \infty$ for all $h \in L^q$ then $f \in L^p$ – GEdgar May 05 '20 at 20:13

Show that if $\int fh < \infty$ for all $h \in L^q$ then $f \in L^p$

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