By Daniel A. Murray
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Extra info for A first course in infinitesimal calculus
If V is a topological vector space and V is its nonstandard hull, then • V is a complete topological vector space. • V is topologically and linearly embedded into V by the map taking each vector v in V to *v + m(0). The closure o f V under this mapping is pns(*V)/m(0 ). • If В is any internal subset o f *V, then В is closed in V. • If A q V, then pns(*A)/m(0) is a completion o f A. The nonstandard hull provides a useful representation theorem for generalized curves. , they are the ele ments of the extension which are infinitesimal at each standard func tion.
Let X be the function defined on T to be alternately 0 and At, and let X be its polygonal extension. In this example, it will be convenient to assume that T has size 2^, where H is infinite. l/H A /V V V V V V V V V \ 2/H Infinitesimal zigzag We will exhibit a sequence of standard polygons whose actions converge to ° [X] by constructing functions similar to X on succes sively coarser timelines, then letting these spill over to timelines with noninfinitesimal increments. In the following, Au will be a multiple of At and Tau will be the set of multiples of Au that lie in T.
A function f : [0,1] - R is Lebesgue measurable iff it has a lifting F :T -*R. • A function f : [0,1] - R is Lebesgue integrable iff it has a lifting F which is S-integrable with respect to p. If F is an S-integrable lifting o f f, then Í fd J[o,i] £ - °i F Jt d p = ° y F(t) At Hermann Rodenhausen gave a useful construction of S-integrable liftings of Lebesgue integrable functions in . Nigel Cutland offered a different proof of Rodenhausen’s lifting theorem in  which used the S-Fubini Theorem implicitly.
A first course in infinitesimal calculus by Daniel A. Murray