Lebesgue measure and general measure theory

1976 ◽  
pp. 38-71
Author(s):  
John J. Benedetto
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
General Measure

Commutative L-algebras and measure theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Measure Theory ◽  
Measurable Space ◽  
Universal Covering ◽  
Continuous Functions ◽  
Stone Space ◽  
Integration Theory ◽  
General Measure ◽  
Lattice Order ◽  
Finitely Additive Measures ◽  
Additive Measures

Abstract Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

scholarly journals Littlewood-Paley theory on Gaussian spaces

1988 ◽  
Vol 109 ◽  
pp. 47-61 ◽  
Author(s):  
Jürgen Potthoff
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Euclidean Spaces ◽  
General Measure ◽  
Finite Dimensional

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

Decidability of the “almost all” theory of degrees

1972 ◽  
Vol 37 (3) ◽  
pp. 501-506 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Random Sequence ◽  
Recursion Theory ◽  
Sharp Contrast ◽  
Standard Theory ◽  
Simple Manner ◽  
Fubini’S Theorem ◽  
Almost All ◽  
Random Pair

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

scholarly journals On Perimeters and Volumes of Fattened Sets

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Andrea C. G. Mennucci
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Geometric Measure Theory ◽  
Coarea Formula ◽  
Compact Set ◽  
Geometric Measure ◽  
General Bound

In this paper we analyze the shape of fattened sets; given a compact set C⊂RN let Cr be its r-fattened set; we prove a general bound rP(Cr)≤NL({Cr∖C}) between the perimeter of Cr and the Lebesgue measure of Cr∖C. We provide two proofs: one elementary and one based on Geometric Measure Theory. Note that, by the Flemin–Rishel coarea formula, P(Cr) is integrable for r∈(0,a). We further show that for any integrable continuous decreasing function ψ:(0,1/2)→(0,∞) there exists a compact set C⊂RN such that P(Cr)≥ψ(r). These results solve a conjecture left open in (Mennucci and Duci, 2015) and provide new insight in applications where the fattened set plays an important role.

Measures: Back and Forth Between Point sets and Large sets

1995 ◽  
Vol 1 (2) ◽  
pp. 170-188 ◽  
Author(s):  
Noa Goldring
Keyword(s):  
Lebesgue Measure ◽  
Set Theory ◽  
Real Line ◽  
Measure Theory ◽  
Impossibility Result ◽  
Point Sets ◽  
Large Sets ◽  
Starting Point ◽  
Standard Set ◽  
Action At A Distance

It was questions about points on the real line that initiated the study of set theory. Points paved the way to point sets and these to ever more abstract sets. And there was more: Reflection on structural properties of point sets not only initiated the study of ordinary sets; it also supplied blueprints for defining extra-ordinary, “large” sets, transcending those provided by standard set theory. In return, the existence of such large sets turned out critical to settling open conjectures about point sets.How to explain such action at a distance between the very large and the rather small? Rather than having an air of magic, could these results rest on deep structural similarities between the two superficially distant species of sets?In this essay I dissect one group of such two-way results. Their linchpin is the notion of measure.§1. Vitali's impossibility result. Our starting point is a problem in measure theory regarding the notion of “Lebesgue measure.” Before presenting the problem, I would like to review the notion of Lebesgue measure. Rather than listing its main properties, I would like to show how Lebesgue measure is born out of an attempt to generalize the notion of the length of an interval to arbitrary sets of reals. One tries to approximate arbitrary sets of reals by intervals, in the hope that the lengths of the intervals will induce a measure on these sets.

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