# Lebesgue measure of two functions.

Assuming Lebesgue Measure

If $f$ and $g$ are measurable functions, show that the set ${\{x:f(x)\gt g(x)\}}$ is measurable.

My idea so far is that since $f$ is measurable, the set ${\{x:f(x)\gt c\}}$ is measurable and likewise for $g$. If $f>g$ a.e then we could say that ${\{x:f(x)\gt g(x)\}}$.

Some definitions:

Measurable function: A function $f$ from $R^n$ into $R$ is said to be a measurable function if, for each $g\in \mathscr L$ such that $g\ge0$, $(-g)\land(f\lor g)$ is integrable.

Measurable Set: A subset $E$ of $R^n$ is measurable if $\chi_E$ is a measurable function.

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It appears that you forgot two "is measurable"s in your question. –  Lord Soth Apr 9 '13 at 18:14
Do you know that linear combinations of measurable functions are measurable? Or are you allowed to use that? –  1015 Apr 9 '13 at 18:23