20 - Differentiation
from PART THREE - LEBESGUE MEASURE AND INTEGRATION
Published online by Cambridge University Press: 05 June 2012
Summary
Lebesgue's Differentiation Theorem
In the last several chapters, we have raised questions about differentiation and about the Fundamental Theorem of Calculus that have yet to be answered. For example:
For which f does the formula hold? If f′ is to be integrable, then at the very least we will need f′ to exist almost everywhere in [a, b]. But this alone is not enough: Recall that the Cantor function f : [0, 1] → [0, 1] satisfies f′ = 0 a.e., but.
Stated in slightly different terms: If g is integrable, is the function differentiable? And, if so, is f′ = g in this case? For which f is it true that for some integrable g?
In our initial discussion of the Stieltjes integral, we briefly considered the problem of finding the density of a thin metal rod with a known distribution of mass. That is, we were handed an increasing function F(x) that gave the mass of that portion of the rod lying on [a, x], and we asked for its density f(x) = F′(x). We side-stepped this question entirely at the time, defining a new integral in the process, but perhaps it merits posing again.
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- Real Analysis , pp. 359 - 378Publisher: Cambridge University PressPrint publication year: 2000