Suppose that $f\colon[a, b] \to \mathbb{R}$ is a function of bounded variation. Show that $f$ is Borel measurable.
I was wondering if I could get a hint.
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6Can you prove that a monotone function is Borel measurable? – Dave L. Renfro Apr 25 '13 at 20:51
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@DaveL.Renfro do you mean that the statement is wrong? – Nov 05 '19 at 12:44
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@Mathstupid: If you can prove that a monotone function is Borel measurable, then a BV function will also be Borel measurable, because every BV measurable function is the algebraic (pointwise) difference of two monotone functions and the difference of two Borel measurable functions is Borel measurable. – Dave L. Renfro Nov 05 '19 at 15:06
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So do you have a source for proving that a monotone function is Borel measurable [email protected] – Intuition Nov 05 '19 at 15:19
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monotone + function + borel + measurable + site:math.stackexchange.com – Dave L. Renfro Nov 05 '19 at 16:06
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You see that for $ f \in BV([a,b]) $ you can write $ f = g - h $ where $ g(x) = V^x_a(f) $ and $ h = g-f $ where both $g $ and $h$ are monotonically increasing hence have countably many discontinuities.
smiley06
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