# Lp space Example

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

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 $$f \not\in L^{\infty}(E)$$.