nLab correspondence between measure and valuation theory

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

Concept↓\Theory→	Measures (traditional)	τ-additive measures	Valuations
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
Finitary linearity	Additivity	Additivity	Modularity
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
Monad*	Giry monad	measure monad on Top	valuation monad on locales?

$*$ 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.

• Steve Vickers, A monad of valuation locales, 2011. Link here.

• Steve Vickers, A localic theory of lower and upper integrals, 2008. Link here.