I am doing a course on Measure theory in my Masters' course.I try to do proofs on my own whenever possible.
Definition
A function $f:X\subset \mathbb {R\to R}$ is called Borel measurable if the inverse image of every Borel set is a Borel, set.
Now,I saw a theorem which I tried to prove on my own:
Th. Every continuous function is Borel measurable.
My proof goes as follows:
It suffices to show that $f^{-1}(a,\infty)$ is a Borel set for each $a\in \mathbb R$.
Now,$f$ is continuous,so $f^{-1}(a,\infty)$ is open in $X$.
So,$f^{-1}(a,\infty)=U\cap X$ for some $U$ open in $X$.
$U$ being open in $\mathbb R$ is a Borel set in $\mathbb R$.
So,$U\cap X$ is a Borel set in $X$.
Thus,$f$ is Borel measurable.
Is the proof ok?The only confusion I have is the meaning of Borel set in $X$ for a subset $X$ of $\mathbb R$,does that mean intersection of $X$ with a Borel set in $\mathbb R$ as I have used here?