Isn't the function φ_n a series of Dirac delta functions? Then, the integrals don't just give zero, but some value of the sum of those points.
@crosseyedcat1183
Күн бұрын
No they're step functions. They're 1 on some interval, and 0 elsewhere. Technically the dirac delta isn't a function and defining it would be done as a measure (or a distribution). We just treat it like a function because it behaves as a function under an integral.
@jeffreyhowarth7850
9 ай бұрын
Are these step functions piecewise functions and is this how we get pointwise convergence?
@alphalunamare
7 күн бұрын
ummmm You don't. If you mean pointwise to mean convergent at every point on the real line then that is not the objective here. The objective is to accept that there may be many non convergent points, but that if that set is 'countable' then the function is still potentially Lebesgue integrable. If I have gotten your question wrong then I apologise :-)
@sinx2247
6 күн бұрын
@@alphalunamare Almost, but there exist null sets (sets with Lebesgue measure 0) that are uncountable. A famous example is the Cantor set
@alphalunamare
6 күн бұрын
@@sinx2247 lol example 1.4.3 in Asplund & Bungart, I remmeber it well.(Page 31). The answer to the original question is also in the examples of Chapter 1. It's 50 years since I last read that chapter so please forgive my apparent ingorance :-)
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