# Are Monotone functions Borel Measurable?

Could you guide me how to prove that any monotone function from $$R\rightarrow R$$ is Borel measurable?

Since monotone functions are continuous away from countably many points, would that be helpful in proving the measurability?

• 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. Commented Dec 6, 2012 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?
– user48405
Commented Dec 6, 2012 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? Commented Dec 6, 2012 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) Commented Dec 6, 2012 at 17:28
• Yes, it completely matches my definition of $\forall B \in \text{Borel Set} \{w: f(w)\in B\} \in F \text{ where F is also Borel Set}$
– user48405
Commented Dec 6, 2012 at 17:30

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