Proving that sum of two measurable functions is measurable.
Solution 1:
This is a method I used in my analysis class. Note that $f(x) + g(x) < t$ iff $f(x) < t-g(x)$ iff there exists a rational number $r$ such that $f(x) < r < t-g(x)$.
Therefore $\{x : f(x) + g(x) < t\} = \bigcup_{r\in\Bbb Q} [f^{-1}((-\infty, r)) \cap g^{-1}((-\infty, t-r))]$.
Both the sets being intersected in the union are measurable sets. Hence the set on the left is also measurable, meaning that $f+g$ is measurable.