A function which is in $L^1$ but does not belong to $L^\infty$ Can someone give me an example of an $L ^1$ function which does not belong to $L^\infty$. In fact we look at $L^1(\Omega,\mathcal{F},P)$, where $(\Omega,\mathcal{F},P)$ denotes a probability space. Of course the function should be unbounded but the integral should exist. Clearly, we can embed $L^\infty$ into $L^1$ in this case. Thank you.

An example for the space $X= (0,1]$ is $\dfrac{1}{\sqrt{x}},$ which has infinite $L^{\infty}$ norm because as $x\to 0^+$ the function $1/\sqrt{x}\to \infty,$ but has $L^1$ norm of $\displaystyle \int^1_0 \frac{1}{\sqrt{x}} dx = 2.$ 
If you want an example for $X=\mathbb{R}$ then $1_{(0,1]} \dfrac{1}{\sqrt{x}}$ works similarly. 
