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?
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2I'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 06 '12 at 17:22
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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? – Dec 06 '12 at 17:23
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1By 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 06 '12 at 17:24
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1If 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 06 '12 at 17:28
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Yes, it completely matches my definition of $\forall B \in \text{Borel Set} {w: f(w)\in B} \in F \text{ where F is also Borel Set}$ – Dec 06 '12 at 17:30
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See this answer: https://math.stackexchange.com/questions/3968451/monotone-functions-are-borel-measurable. – shark Jun 07 '23 at 23:26
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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 $f^{-1}((-\infty,x])$.
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