I want to prove the following statement:
If $f$ is a real function over a measure space $X$ such {$x$/ $f(x)$ $\ge$ $r$} is measurable for each rational , then $f$ is measurable.
So far , this is what I have:
Picking up a rational number sequence {$q_{1}$$>$$q_{2}$...$>$$q_{n}$$>$$r$} then lim {$q_{n}$}=$r$ as te sequence {$q_{n}$} is decreasing when $n$ is incresing and is also lower bounded by $r$ then this implies that ($r$ ,$\infty$)=$\bigcup_{n=1}^{\infty}$ [$q_{n}$ ,$\infty$) and I know that a sigma algebra is closed over countable unions
How can I end up this proof and properly argue it in case my thought so far is OK.??
Thanks