Non-Standard Integration

The most vivid vindication of the physicist's calculus. This page of the NSA spine defines the integral literally as Leibniz did — as a sum of infinitely many infinitesimally thin slices $fdx$ — realized as a hyperfinite sum (internal-sets.md §4) and rounded to a real by the standard part (hyperreals.md §4). We then prove it equals the Darboux/Riemann integral of riemann-integral.md, so "$\int =\sum fdx$" is not a heuristic but a theorem.

References: Keisler, Elementary Calculus, ch. 4 & 6; Goldblatt, Lectures on the Hyperreals, ch. 9; Robinson, Non-standard Analysis, ch. 4.

1. Hyperfinite partitions and the infinitesimal slice

Let $f:[a,b]\to \mathbb{R}$ be bounded, and fix an unlimited $H\in {}^{\ast }\mathbb{N}$ (internal-sets.md §1). Partition $[a,b]$ into $H$ pieces of infinitesimal width

$$\Delta x=\frac{b-a}{H}\approx 0,x_i=a+i\Delta x(i=0,1,\dots ,H).$$

This is an internal, hyperfinite partition: by transfer it has all the properties of an ordinary equal-spacing partition (riemann-integral.md §1), except its mesh is infinitesimal and its number of subintervals is infinite. The single slice over $[x_{i-1},x_i]$ has area $\approx {}^{\ast }f(x_i)\Delta x$ — Leibniz's "$fdx$."

2. The integral as the standard part of a hyperfinite sum

Definition. The hyperfinite Riemann sum of $f$ over the grid is the (single, exact) hyperreal $$S_H(f)=\sum _{i=1}^H{}^{\ast }f(x_i)\Delta x,$$ defined by transfer of the finite-sum recursion (internal-sets.md §4). If $S_H(f)$ is finite and its standard part is the same real for every unlimited $H$ (and every choice of evaluation point $x_i^{\ast }\in [x_{i-1},x_i]$), then $f$ is integrable and $${\int }_a^{b}fdx:=\mathrm{st}(S_H(f)).$$

This is exactly the mental image behind "chop into slices, add them up, let the slices get infinitely thin": the summation is genuine (hyperfinite), the slices are genuinely infinitesimal, and the only limiting step is the final $\mathrm{st}$.

3. Equivalence with the Darboux/Riemann integral

Theorem. For bounded $f$ on $[a,b]$, $f$ is integrable in the sense of §2 iff it is Riemann/Darboux integrable (riemann-integral.md §2), and the two values agree.

Proof sketch. Let $L(f,P)\le S\le U(f,P)$ be the lower/upper Darboux sums of a standard equal partition $P$ with $n$ pieces (riemann-integral.md §1). The sentence "$L(f,P_n)\le {\sum }_{i}f(x_i)\Delta x_i\le U(f,P_n)$ for all $n$" transfers to all hypernaturals, so for unlimited $H$,

$${}^{\ast }\underline{\int }\le {}^{\ast }L(f,P_H)\le S_H(f)\le {}^{\ast }U(f,P_H)\le {}^{\ast }\overline{\int }.$$

If $f$ is Darboux integrable, $\overline{\int }=\underline{\int }=I$ real, and by transfer ${}^{\ast }U(f,P_H)-{}^{\ast }L(f,P_H)\approx 0$ (the mesh is infinitesimal), squeezing $S_H(f)\approx I$; hence $\mathrm{st}(S_H(f))=I$ independent of $H$. Conversely, if $\mathrm{st}(S_H(f))$ is a single real for all unlimited $H$, overspill (internal-sets.md §5) forces the standard upper/lower sums to converge, giving Darboux integrability. $\square$

Corollary (continuous $\Rightarrow$ integrable, nonstandardly). If $f$ is continuous on $[a,b]$ it is uniformly microcontinuous there (infinitesimal-calculus.md §2), so on the infinitesimal-mesh grid $M_i-m_i\approx 0$ uniformly, whence ${}^{\ast }U-{}^{\ast }>L\approx 0$ and $f$ is integrable. This is the nonstandard face of the classical uniform-continuity proof (riemann-integral.md §3).

4. The Fundamental Theorem of Calculus

The FTC is especially transparent infinitesimally. Let $F(x)={\int }_a^{x}f$ with $f$ continuous, and $\Delta x\ne 0$ infinitesimal. The increment $F(x+\Delta x)-F(x)$ is the area of one infinitesimal slice, so by microcontinuity of $f$,

$$F(x+\Delta x)-F(x)={\int }_x^{x+\Delta x}f={}^{\ast }f(\xi )\Delta x\text{for some }\xi \approx x,$$

(the hyperfinite mean-value/pinching step), and therefore

$$\frac{F(x+\Delta x)-F(x)}{\Delta x}={}^{\ast }f(\xi )\approx f(x),\text{so}{F}^{\prime }(x)=\mathrm{st}({}^{\ast }f(\xi ))=f(x).$$

That is $\frac{d}{dx}{\int }_a^{x}f=f(x)$ — the first form of the FTC (riemann-integral.md) — obtained by dividing an infinitesimal area by an infinitesimal width and taking the shadow. The second form ${\int }_a^b{F}^{\prime }=F(b)-F(a)$ follows by transfer of the telescoping identity ${\sum }_i(F(x_i)-F(x_{i-1}))=F(b)-F(a)$ across the hyperfinite grid.

5. Substitution and the Leibniz notation, justified

The change-of-variables rule "$u=g(x)$, $du=g^{\prime }(x)dx$" is now honest bookkeeping. On the hyperfinite grid, $\Delta u_i={}^{\ast }g(x_i)-{}^{\ast }g(x_{i-1})={}^{\ast }{g}^{\prime }({\xi }_i)\Delta x$ for ${\xi }_i\approx x_i$ (the FTC step of §4 applied to $g$), so

$$\sum _i{}^{\ast }f({}^{\ast }g(x_i))\Delta u_i=\sum _i{}^{\ast }f({}^{\ast }g(x_i)){}^{\ast }{g}^{\prime }({\xi }_i)\Delta x\approx \sum _i{}^{\ast }(f\circ g)(x_i){}^{\ast }{g}^{\prime }(x_i)\Delta x,$$

and taking standard parts yields ${\int }_{g(a)}^{g(b)}f(u)du={\int }_a^{b}f(g(x))g^{\prime }(x)dx$ (riemann-integral.md). The "$du=g^{\prime }(x)dx$" that physicists cancel is the exact statement $\Delta u_i\approx {}^{\ast }{g}^{\prime }(x_i)\Delta x$ between infinitesimals.

6. The reach of the method: Loeb measure (pointer)

The hyperfinite-sum idea extends far beyond the Riemann integral. Applying the standard part to the internal counting measure on a hyperfinite set and then completing produces the Loeb measure — a genuine, countably additive (standard) measure. Loeb measures give nonstandard constructions of Lebesgue measure, Brownian motion, and more, and are the bridge from NSA to full measure theory. That development is beyond this spine (a possible future measure-theory.md); we record only that "sum of infinitesimals" scales all the way up to Lebesgue integration.

Where this page is used

• Completes the infinitesimal reconstruction of calculus begun in infinitesimal-calculus.md.
• Is the literal rigorous meaning of "$\int fdx$ = sum of slices" claimed in physics-use-of-calculus.md §2 & §4.1.
• Recovers the Riemann integral of riemann-integral.md §2.1.

Next: Sequences, Limits, and Topology, Non-Standardly — convergence and compactness through the halo.