Additive Gaussian white noise - bandwidth
Just to be clear about the units:

\begin{align}
\Omega &amp;= 2\pi F, \quad \text{with $\Omega$ in } \rm\underline{radians/sec}\text{ and $F$ in } \rm\underline{cycles/sec} \text{ or }\rm\underline{Hz}\tag{1}\\
\omega &amp;= 2\pi f, \quad \text{with $\omega$ in } \rm\underline{radians/samples}\text{ and $f$ in } \rm\underline{cycle/sample}\tag{2}\\
\omega&amp;=\Omega T_s\quad \text{and}\quad f=F/F_s\tag{3}
\end{align}
Read @Dilip Sarwate's comments; what should happen there is a change of variables. 
\begin{align}
\omega &amp;= 2\pi f\Longrightarrow d\omega = 2\pi df\iff df=\frac{d\omega}{2\pi}\tag{4}\\
f&amp;=\pm B \Longrightarrow \omega= \pm 2\pi B\tag{5}\\
\end{align}
Using $(4)$ and $(5)$ you then have the following results:
$$
P_0=\int_{-B}^{B} \frac N2 df = NB \Longrightarrow P_0= \int_{-2\pi B}^{2\pi B} \frac N2 \cdot \frac{1}{2\pi}d\omega=\frac{1}{2\pi}\int_{-2\pi B}^{2\pi B} \frac N2 d\omega = NB\tag{6}\\
$$