Lebesgue integrable function and square-integrable functions?

Question

I heard that lebesgue-integrable function is square-integrable function, and vice versa. Why is this the case? (in my textbook) On the definition of Lebesgue integration, it only defines that function, when integrated with some measure is less than infinity, and nothing about square-integrable stuffs, so I ask here.

Answer 1

This is false; for example, let $f_n (x) = n^{3/2}$ for $0\le x \le 1/n^2$ and $0$ otherwise. Then $||f_n||_1 = n^{-1/2} \to 0$ as $n\to \infty$ while $||f_n||_2 = n^{1/2} \to \infty$ as $n\to \infty.$ The point here is that the (pointwise) limit of measurable functions is measurable, so taking the limit gives a function $f\in L^{1}$ but not in $L^{2}.$ In general, $L^{p}\not\subseteq L^{q}$ and $L^{q} \not\subseteq L^{p}$ for $p\not= q.$