nLab correspondence between measure and valuation theory

Contents

Context

Measure and probability theory

measure theory

probability theory

Contents

Idea

Valuations and measures have very similar constructions and applications, with $\tau$-additive measures somewhat in between.

Here we review which concepts correspond to each other. Note that on many topological spaces of interest, such as metric spaces, most of the constructions below coincide.

Table

Spaces Measurable spaces Topological spaces Locales
Maps Measurable maps Continuous maps Continuous maps (of locales)
Events Measurable sets Open sets Open sets (elements of the frame)
Values Non-negative reals Non-negative reals, with the lower semicontinuous topology Non-negative lower reals
Functionals Measures τ-additive Borel measures Continuous valuations
Infinitary linearity $\sigma$-additivity $\tau$-additivity (Scott) continuity
Integrands Measurable real-valued functions Lower semicontinuous real functions Continuous functions into the lower reals
Approximation of integrands Pointwise-increasing sequences (of measurable real functions) Pointwise-increasing nets (of lower semicontinuous real functions) Pointwise-increasing nets (of continuous functions into the lower reals)
Integral Lebesgue integral Lebesgue integral Lower integral

$*$ See also monads of probability, measures, and valuations.

References

• V. Bogachev, Measure Theory, vol. 2 (2007).

• Reinhold Heckmann, Spaces of valuations, Papers on General Topology and Ap-plications, 1996. Link here.

• Mauricio Alvarez-Manilla, Achin Jung, Klaus Keimel, The probabilistic powerdomain for stably compact spaces, Theoretical Computer Science 328, 2004. Link here.

• Olaf Kirch, Bereiche und Bewertungen (in German), Master Thesis, Technische Hochschule Darmstadt, 1993. Link here.

• Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004. Link here

• Thierry Coquand and Bas Spitters, Integrals and Valuations, 2009. Link here.