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Could you guide me how to prove that any monotone function from $R\rightarrow R$ is Borel measurable? Should we separate the functions into continuous and non-continuous? How to prove for not continuous points?

Thanks for your help

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I'd try to apply the definition directly. That is, try to show that sets of the form $\{x\in \mathbb{R}\ |\ F(x)\ge t\}$ are Borel. –  Giuseppe Negro Dec 6 '12 at 17:22
Hi @GiuseppeNegro, I actually have hard time understanding this method. I always use the basic definition of looking at pre-image. Could you explain this a little more. How we show/use this? –  Eli Dec 6 '12 at 17:23
By definition, a function $\mathbb{R}\to \mathbb{R}$ is Borel-measurable when the preimages of open subsets of $\mathbb{R}$ are Borel sets of $\mathbb{R}$. Do you agree with this definition? –  Giuseppe Negro Dec 6 '12 at 17:24
If you agree, then you can convince yourself that, actually, it is enough to check that the preimages of half-lines are Borel. More precisely, $F\colon \mathbb{R}\to \mathbb{R}$ is Borel measurable if and only if for every $t \in \mathbb{R}$ the following set is Borel: $$\{x\in \mathbb{R}\ |\ F(x)\ge t\}$$ (Cfr. Rudin, Real and complex analysis, 3rd ed., Theorem 1.12) –  Giuseppe Negro Dec 6 '12 at 17:28
Yes, it completely matches my definition of $\forall B \in \text{Borel Set} \{w: f(w)\in B\} \in F \text{ where F is also Borel Set}$ –  Eli Dec 6 '12 at 17:30
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1 Answer

Hint: If $f$ is monotone, then for every real number $x$, the set $f^{-1}((-\infty,x])=\{t\mid f(t)\leqslant x\}$ is $\varnothing$ or $(-\infty,z)$ or $(-\infty,z]$ or $(z,+\infty)$ or $[z,+\infty)$ for some real number $z$.

To show this, assume for example that $f$ is nondecreasing and that $u$ is in $f^{-1}((-\infty,x])$, then show that $v$ is in $f^{-1}((-\infty,x])$ for every $v\leqslant u$.

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