By V. I. Smirnov, A. J. Lohwater
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Extra info for A Course of Higher Mathematics. Integration and Functional Analysis
With the subdivision fixed above, we can write (70) in the form where | Θ | < 1. Similar arguments lead us to the formula where | Θη | < 1. Subtraction term by term gives us The points x^ are fixed, and, since the gn(x) converge to g(x) at these points, the sum appearing in the last formula has an absolute value less than ε for all sufficiently large n. The last formula thus leads us to the inequality, for these n: whence (69) follows, since ε is arbitrary. Note. Suppose that the gn(x) tend to g(x) only on a set of points dense in [a, 6], instead of at every point, both ends of the interval being included in the set; the limit function g(x) is assumed to be of bounded variation.
A similar treatment can be given of the third term on the right-hand side of (72). Hence, given any positive e, we can fix a and b of the above-mentioned set, everywhere dense in [— oo, + o o ] , such that the first and third terms on the right of ( 72) are less than e for all sufficiently large n. e. the second term on the right of (72) is also less than ε for all sufficiently large n. Thus the left-hand side of (72) is less than 3ε for all sufficiently large n, whence (71) follows, in view of the arbitrariness of ε.
We shall now prove a similar theorem for the case when the fn(x) are unbounded in [— oo, + oo], and the integral over this interval has to be understood as an improper integral. THEOREM 2. Let fn(x) be continuous inside [— oo, + oo], let the improper integrals +~ J fn(x)dg(x)= — oo β lim j fn(x) dg(x) a—►— oe a (64) ß-++o. exist uniformly with respect to n, fn(x) -+ f(x) uniformly in any finite interval and g(x) be a function of bounded variation in any finite interval. Nowy the (improper) integral of f(x) with respect to g(x) over the interval [— oo, + o o ] exists, and (63) holds.
A Course of Higher Mathematics. Integration and Functional Analysis by V. I. Smirnov, A. J. Lohwater