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Let $f:\mathbb{R}\to \mathbb{R}$ be a function of bounded variation. Define $\overline{f}(x)$ by $$\overline{f}(x) = \limsup_{\mu(I)\to 0} \frac{1}{\mu(I)} \int_I f(x) dx,$$ where $I$ ranges over intervals containing $x$ and $\mu$ is the Lebesgue measure. It is known that the total variation $|\overline{f}|_{TV}$ of $\overline{f}$ is minimal among those of all functions equal to $f$ almost everywhere; see, e.g., Lemma 3.3 in https://arxiv.org/pdf/math/0601044.pdf.

Now, that should be a very classical result. Does anybody have an older or standard reference?

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  • $\begingroup$ $I$ is an interval, as I said clearly. $\mu$ is the standard Lebesgue measure. $\endgroup$
    – H A Helfgott
    Commented May 1, 2020 at 15:29
  • $\begingroup$ It doesn't matter whether you let $I$ range over open or closed intervals. $\endgroup$
    – H A Helfgott
    Commented May 1, 2020 at 16:10
  • $\begingroup$ Oh as $I$ dimnishes, its still always be a neighbourhood of $x$? $\endgroup$
    – Rajesh D
    Commented May 2, 2020 at 2:16
  • $\begingroup$ Yes: "$I$ ranges over intervals containing $x$". $\endgroup$
    – H A Helfgott
    Commented May 2, 2020 at 6:20

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