Example showing L1 is not a reflexive space

Question

I know that the $L^p$ spaces are reflexive for $1<p<\infty$. I want to explicitly show that $L^1((0,1),\mathbb{R})$ is not reflexive by finding an element of $L^1$** that is not in $L^1$. To be more precise:

There is a canonical embedding $J: X \rightarrow X^{**}$ that is given by

$$J(x)(y)=y(x)$$

I want to find an element of $L^1$** that doesn't get mapped to by $J$.

Answer 1

This was answered in this question, the top two answers have a textbook reference and an actual proof.