If f and g are 2 measurable functions defined on a measurable subset E of R, then show that
$\displaystyle E_{1}=\{ x \in E : f(x) < g(x)\} $ is also measurable and
$\displaystyle E_{2} = \{ x \in E : f(x) = g(x)\} $ is measurable
Let h:=f-g. Then your second problem is equivalent to showing $\displaystyle \{ x\in E : h(x) = 0\}$ is measurable. But this is $\displaystyle h^{-1}(0)$ and since h is measurable, and {0} is measurable, $\displaystyle E_2$ is too. Use similar ideas for $\displaystyle E_1$.