Lp space Example

Question

How are spaces connected $L_{\infty}(E)$ and $L_{p}(E)$, $|E| = \infty$?

$$f \in L_\infty, \text{ but } f\notin L_1 \quad f = \frac{1}{x} \quad E = [1, \infty)$$

What can we say about reverse insertion? I think there is a suitable example

Answer 1

Neither containment holds for Lebesgue measure if $|E| = \infty$. Your example shows that $L^{\infty}(E) \not\subset L^1(E)$. For a counterexample in the reverse direction, let $E = (0,\infty)$ and define $$f(x) = \begin{cases} \frac{1}{\sqrt{x}} & \text{ if }0 < x < 1 \\ 0 & \text{ otherwise} \end{cases}$$ Then $f \in L^1(E)$ but $f \not\in L^{\infty}(E)$.