Suppose I have the probability space $(X,\mathcal{E},P)$ and a random variable $u : X \to \mathbb{R}$. $u$ induces a probability $Q (B) = P (u^{-1}(B))~\forall B \in \mathcal{B}$, the Borel $\sigma$-algebra on $\mathbb{R}$ and we can consider a new probability space $(\mathbb{R},\mathcal{B},Q)$.
Now, I am having trouble connecting this theory to practical applications. For example, can I let $u$ to be a Standard Normal Random Variable? If so, the expectation of $u$ is defined as $E[u] = \int\limits_X u dP$. How does this equal $\int\limits_{\mathbb{R}} x f(x) dx$ where $f(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$?
I guess that the induced measure has to be brought into play along with some Radon-Nikodym theorem to show this but I am not sure how to do so.
Any help is greatly appreciated.
Thanks, Phanindra