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 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
where 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.