Integrals are not infinite sums

I realized a mistake I’ve had today while sitting in my local analysis class. Realized is generous, rather it was pointed out to me.

I’d generally thought of integrals such as \int_a^b f(x) dx as sums of f(x) where the index ranged over [a,b]. Clearly I had never thought very hard about this. If you add up every value of f(x) and it’s not zero a lot your sum will diverge. The integral is different because of the measure it uses (I’m assuming a Lesbesgue measure on R). The measure for the uncountable sum is the counting measure on the set [a,b] in which every singleton has measure one, versus Lebesgue which gives measure zero to such a set.

We can certainly write something like

\sum_{\alpha\in A} f(\alpha)=\int_A f(\alpha) d\mu

where \mu is the counting measure, but we can’t think of it as the Lebesgue measure. Which means I think it’s about time that I look into the Stieltjes integral and see how to understand the spectrum of an operator that has both discrete and continuous spectrum. Next post perhaps.


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