Measure theory in practice I am trying to unite my knowledge of statistics and measure theory by considering the following example. 
Suppose we have a measurable space $(\Omega_1,B_1)$ and a random variable (measurable function) on the space, call it $X$: $\Omega_1 \rightarrow R$.
Suppose we know the distribution function of $X$, say it is normal $X \sim N(0,1)$. Now consider the random variable 
$$Y=X +5$$
We know from basic statistics that $Y\sim (5,1)$, but how can we prove that by using the definition of $X$ and composition of functions? Explanations that are step by step greatly appreciated! 

As S. van Nigtevecht says, when you say $X\sim N(0,1)$ you are saying there is a probability measure added to the setup, so we have in all a measure space $(\Omega, B, P)$, and a pair of functions $X$ and $Y$ such that $Y(\omega)=X(\omega)+5$ for all $\omega\in\Omega$.  If you want, $Y$ is the composition of $X$ with the "add to 5" function.  We also  know also that $P(\{\omega: X(\omega)\le a\})=\int_{-\infty}^a \phi(t)\,dt$, where $\phi$ is the $N(0,1)$ density function.  So $Y^{-1}([-\infty,x]) = X^{-1}([-\infty,x+5]),$ and  $$P(Y\le x) = P(X\le x+5 ) = P(\{\omega: X(\omega)\le x+5\}) = \int_{-\infty}^{x+5}\phi(t)\,dt$$ and so on.  (I have dropped the subscript $1$ here & there.)
Is this what you want?
