Titlebook: Measure and Integral; Volume 1 John L. Kelley,T. P. Srinivasan Textbook 1988 Springer-Verlag New York Inc. 1988 banach spaces.convergence.i

是限制

分期付款

Pre-Measures,We consider briefly the class of length functions. These will turn out to be precisely the functions on the family of closed intervals that can be extended to become measures; these are examples of pre-measures. Their theory furnishes a concrete illustration of the general construction of measures.

弄皱

Pre-Integral to Integral,This section is devoted to the construction of an integral from a pre-integral, and to a few consequences. Among these consequences are norm completeness, Fatou’s lemma, the monotone convergence theorem and the dominated convergence theorem for an arbitrary integral.

Reservation

The Integral , on ,(,),This section is devoted to the construction of an integral .from a measure ., to the relationships between . and . (especially for Borel measures . for ℝ), and to a brief consideration of the vector spaces ., 1 ≦ . ≦∞, associated with ..

FLACK

Ligament

Pre-Measure to Pre-Integral,other way: does the function χ. ↦.[.] have a linear extension to the vector space of linear combinations of functions of the form χ.? It turns out that this is the case, and that it is a consequence of the fact that λ has an additive extension to a ring of sets containing the closed intervals, as we presently demonstrate.

眼界

Measures* and Mappings, the prototypical example of a measure*. A function . is . (or . . on . iff it is integrable (integrable*) w.r.t. the measure . . . < ∞} and in this case ∫ . = ∫ .. Thus the integral w.r.t. classical Lebesgue measure is indentical with the integral w.r.t. Λ..

medieval