We have a non-negative random variable $z$. We apply a simple location/scale transformation with $a$ and $b$ being the transformation parameters, respectively.
I am trying to show that this change-of-variables argument is true for any function $f$
$$E_{\text{base}}f(a+bz)=E_{\text{transformed}}f(z),$$
where $E$ is the expected value under the base and transformed distributions.
I started with the definition of the expected value $$E_{\text{base}}f(a+bz)=\int f(a+bz)P_{\text{base}}(z)dz$$ and $$E_{\text{transformed}}f(z)=\int f(z)P_{\text{transformed}}(z)dz$$
If we define $u=a+bz$, then we can write $$E_{\text{base}}f(u)=\int f(u)P_{\text{base}}(\frac{u-a}{b})\frac{1}{b}du$$
But now what?
Hint: You are done if you can prove, that $$P_{\text{transformed}}(u) = P_{\text{base}}(\frac{u-a}{b})\frac{1}{b}.$$