The definition of integration in non-standard analysis is $$\int_a^b f(x)dx:= st\left(\sum_a^b f(x)dx \right),$$ as given by Keisler (you could also use the transfer principle for internal functions). This sum involves adding up infinitely many numbers, multiplying by an infinitesimal, then taking the standard part. My question is how the infinite sum is guaranteed to converge for an integrable function. As the sequences of partial sums aren’t usually Cauchy (and even if they were, the hyperreals aren’t complete), what notion of convergence is being used here to define integration?
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These "infinite sums" are not defined using any notion of convergence. Instead, they are just a certain internal function. Specifically, there is a function $S:\mathbb{R}_+\to\mathbb{R}$ defined by $$S(h)=\sum_{k=0}^{\lfloor(b-a)/h\rfloor}f(a+kh)h$$ (in other words, a sum of terms $f(x)h$ where $x$ starts at $a$ and goes up by steps of $h$ until it passes $b$). This then extends to an internal function ${}^*\mathbb{R}_+\to{}^*\mathbb{R}$ and "$\sum_{a}^bf(x)dx$" is the value of this internal function on an infinitesimal $dx$.
More generally, given an internal function $f$ and a hypernatural number $N$, you can define a "hyperfinite sum" $\sum_{k=0}^Nf(k)$. This is just defined as the internal extension of the operation which takes an ordinary function $f$ and an ordinary natural number $N$ and computes the sum $\sum_{k=0}^Nf(k)$.
Eric Wofsey
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