Let $(R,L, λ)$ be the Lebesgue measure space and $f : R → R$ a function with $f(x) = 127e^x$ for all $x ∈ Q^c$. Show that $f$ is measurable with respect to $L$.
This is a question on a past exam paper for a measure theory course I'm taking and it has me stumped.
I'm thinking I might need to bring in a function $g$ and then prove that since $g$ is measurable then so is $f$ but I'm not quite sure. A push in the right direction would be much appreciated. Thank you in advance.
Assuming that $Q$ is the set of rational numbers, you are given that $f$ is equal almost everywhere to a continuous function, and is therefore measurable.
Can you prove that if $g$ is measurable, $f$ is a function, and $\lambda(\{f \not= g\}) = 0$, then $f$ is measurable too?