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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.2012-12-06
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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?2012-12-06
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    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?2012-12-06
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    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)2012-12-06
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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}$2012-12-06

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