By T.H. Hildebrandt

ISBN-10: 0080873251

ISBN-13: 9780080873251

ISBN-10: 0123480507

ISBN-13: 9780123480507

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Additional info for Introduction to Theory of Integration (Pure & Applied Mathematics)

Example text

Definition. In a linear normed space, a linear functional is continuous if lim,z11 x, - x 11 = 0 implies lim,f(x,) = f(x) or lim,f(x, - x) = 0. 8 THEOREM. A necessary and sufficient condition that a linear functional on a linear space X be continuous is that there exists a constant M such that 1 f ( x ) I 5 M 11 x 1 I for all x of X , or that f(x) be bounded (or limited) on X . 3. This Page Intentionally Left Blank CHAPTER I I RIEMANNIAN T Y P E OF INTEGRATION In this chapter, we take up some types of integrals suggested by the Riemann definition of integral.

A monotonic nondecreasing (nonincreasing) function is obviously of bounded variation with J: I df I = I f ( b ) - f ( a ) I. As a consequence any linear combination of monotonic nondecreasing functions Etatf,(x) is also of bounded variation. 2. THEOREM. Every function of bounded variation on a linear interval [a, b] can be expressed as the difference of two monotonic nondecreasing functions. For if v(x) = J: 1 df 1, then v(x) (f(x) - f ( a ) ) and v(x) (f(x) - f ( a ) ) are both monotone nondecreasing.

The importance of the pseudoadditive condition is that it is the connecting link between the norm and a-integrals. 10. T H E O R E M . A necessary and sufficient condition that the norm integral of an interval function f ( I ) exists is that the a-integral exist and that f ( I ) be pseudoadditive at every interior point of [a, b]. Since a, 2 u 2 implies that I crl I 5 I u 2 1, it follows that if the norm integral exists, the a-integral exists and the values agree. Conversely, suppose a J]:,f(dI)exists.