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.
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.
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.
Last revised on October 20, 2019 at 22:00:10. See the history of this page for a list of all contributions to it.