是限制
发表于 2025-3-28 15:59:13
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分期付款
发表于 2025-3-28 20:42:25
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
发表于 2025-3-29 01:27:13
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
发表于 2025-3-29 04:00:11
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
发表于 2025-3-29 09:40:19
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Ligament
发表于 2025-3-29 14:14:59
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
发表于 2025-3-29 17:14:36
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
发表于 2025-3-29 22:30:19
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