The bookUnit 1 · Lebesgue Measure and Integration20 / 83

L^p spaces essentials

Why this is asked: L^p inclusions go one way on finite measure spaces and the other way for ℓ^p — getting the direction right is most of the battle.

L^p spaces: inequalities and inclusions

The written notes for this page come with the Notes pack. The video, the visual and the practice below are free.

See it move

The inclusion reverses between a probability space and the real line. Both directions, with the counterexample for each.

The trap here

“ $L^{1}[0,1] \subseteq L^{2}[0,1]$ ” — false

$f(x) = 1/\sqrt{x}$

$\int{}f = 2 < \infty$ but $\int{}f^{2} = \int$ dx $/x = \infty$ . On a finite measure space the inclusion runs the other way.