Loeb space
In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using non-standard analysis.
Construction
Loeb's construction starts with a finitely additive map ν from an internal algebra A of sets to the non-standard reals. Define μ to be given by the standard part of ν, so that μ is a finitely additive map from A to the extended reals R∪∞∪–∞. Even if A is a non-standard σ-algebra, the algebra A need not be an ordinary σ-algebra as it is not usually closed under countable unions. Instead the algebra A has the property that if a set in it is the union of a countable family of elements of A, then the set is the union of a finite number of elements of the family, so in particular any finitely additive map (such as μ) from A to the extended reals is automatically countably additive. Define M to be the σ-algebra generated by A. Then by Carathéodory's extension theorem the measure μ on A extends to a countably additive measure on M, called a Loeb measure.
References
Measure
Measure may refer to:
Legal
Mathematics and science
Units
Measure (album)
Measure is the second album from Matt Pond PA, released in 2000.
Track listing
Personnel
Technical personnel
Termination analysis
In computer science, a termination analysis is program analysis which attempts to determine whether the evaluation of a given program will definitely terminate. Because the halting problem is undecidable, termination analysis cannot be total. The aim is to find the answer "program does terminate" (or "program does not terminate") whenever this is possible. Without success the algorithm (or human) working on the termination analysis may answer with "maybe" or continue working infinitely long.
Termination proof
A termination proof is a type of mathematical proof that plays a critical role in formal verification because total correctness of an algorithm depends on termination.
A simple, general method for constructing termination proofs involves associating a measure with each step of an algorithm. The measure is taken from the domain of a well-founded relation, such as from the ordinal numbers. If the measure "decreases" according to the relation along every possible step of the algorithm, it must terminate, because there are no infinite descending chains with respect to a well-founded relation.
Podcasts:
