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I'm sorry, but my measure theory is practically non-existent. I'm looking at this transformation: $$T : L^1(\mathbb{R}) \rightarrow \mathbb{R}^\mathbb{R}$$ $$\begin{aligned} (Tf)(x)&:=\mu\{ y: |y - x| \le \tfrac12{f(y)},\enspace f(y)>0 \}\\ &\quad- \mu\{ y : |y - x| \le -\tfrac12{f(y)},\enspace f(y)<0 \}\end{aligned} $$ where $\mu$ is the borel measure. Essentially, what I'm trying to construct here is a transformation that, in a wishy-washy sense, takes all the ideal, infinitely thin rectangles that form an ideal Riemann sum, turn them on their side (so they have infinitely small height), and taking the transformed function value to be the total height of all these infinitely many infinitely thin rectangles. I know it's not very rigorous, but there's no issue (as far as I can tell) with the actual definition.

Here's what I'm looking for:

  • Confirmation that $Tf$ is integrable, and preferably, that $\int Tf d\mu = \int f d\mu$.

  • A name for this transformation (if one exists)

  • A simpler representation of this transformation (if one exists)

Any help would be appreciated. Thanks!

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The definition only came out as [Math processing error] for me, so I tweaked the TeX until it worked. – Harald Hanche-Olsen Oct 4 '12 at 13:36

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