Q:

Question about the notation or idea used in this proof

Could someone please explain the idea behind the red arrow. Not sure if I follow. To me it looks like treating an indefinite integral as a definite integral or am I barking up the wrong acyclic connected graph? 

Thanks

A:

In general, for a differentiable function $\;f\;$ , we have that
$$\int f'(x)\,e^{f(x)}\,dx= e^{f(x)}+C$$
Since you have an improper integral, you get
$$\int\limits_{-\infty}^\infty f'(z)\,e^{f(z)}\,dz=\left. e^{f(z)}\right|_{-\infty}^\infty\;,\;\;\text{with}\;\;f(z)=-\frac12\frac{z^2}{\sigma^2}$$
and, in fact, what one must do is
$$\left.e^{f(x)}\right|_{-\infty}^\infty=\lim_{x\to\infty}e^{f(x)}-\lim_{x\to-\infty}e^{f(x)}$$

