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! 

In order to properly speak of a random variable's distribution, you need a measure space (with a measure), and specifically a probability space $(\Omega, \mathscr A, \mathbb P)$, where of course $\mathbb P (\Omega) = 1$.
The distribution of a random variable $X$ is the image measure $X(\mathbb P)$, i.e.
$$ X(\mathbb P)(A) :=\mathbb P(X^{-1}(A)) \quad \text{for }A\in\mathscr A,$$
Outside of the measure-theory context, I first encountered this as the 'cumulative distribution function' of a random variable, i.e. the function that gives the probability of $X$ being less than or equal to $x$:
$$ F_X(x) = \mathbb P (X \leq x) = \mathbb P \left(X^{-1} (-\infty, x]\right).$$
You may recognise $(-\infty, x]$ as being the sets that generate the Borel sets on $\mathbb R$.
A random variable has normal distribution $N(0,1)$ when this image measure is given by:
$$ X(\mathbb P)(A) = \int_A e^{-x^2/2} \, \mathrm \lambda(\mathrm dx),$$
where $\lambda$ is the one dimensional Lebesgue-measure. Phrased in less measure-theory heavy terms,
$$ F_X(x) = \int_{-\infty}^x e^{t^2/2} \, \mathrm dt.$$
If $Y = X+5$, then we have $Y(\mathbb P)(A)=X(\mathbb P)(A-5)$, giving
$$ Y(\mathbb P)(A) = \int_{A-5} e^{x^2/2} \, \lambda(\mathrm dx)\\
=\int_A e^{(x-5)^2/2} \, \lambda(\mathrm dx),$$
which is the definition of $Y$ having $N(5,1)$ distribution. Here we used the fact that $x \mapsto x +5$ is measurable (because it is continuous) to justify the last step.
