Abstract: In this paper, a class of approximate interpolation neural networks is constructed to approximate Lebesgue integrable functions. It is showed that the networks can arbitrarily approximate ...

In most texts, [1] (p. 261), [2] (p. 187), [3] (p. 129), to name a few, the Density Theorem is proven using the fact that if f is Lebesgue integrable and if σ = òf, then σ¢ = f a.e. This result in ...

The Lebesgue-Nikodym Theorem states that for a Lebesgue measure an additive set function which is -absolutely continuous is the integral of a Lebegsue integrable a measurable function ; that is, for ...

Using functions of bounded variation, we define a Volterra type derivative of the linear functional associated with a Lebesgue integrable function and show that it is equal to this function almost ...

I’ve just arXived a new paper: The categorical origins of Lebesgue integration (arXiv:2011.00412). Longtime Café readers may remember that I blogged about this stuff back in 2014, but I’ve only just ...

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I’ve just come back from the big annual-ish category theory meeting, Category Theory 2014 in Cambridge, also attended by Café hosts Emily and Simon. The talk I gave there was called The categorical ...

The Monthly publishes articles, as well as notes and other features, about mathematics and the profession. Its readers span a broad spectrum of mathematical interests, and include professional ...