Extension of measures: a categorical approach

@article{Frič2005,
abstract = {We present a categorical approach to the extension of probabilities, i.e. normed $\sigma $-additive measures. J. Novák showed that each bounded $\sigma $-additive measure on a ring of sets $\mathbb \{A\}$ is sequentially continuous and pointed out the topological aspects of the extension of such measures on $\mathbb \{A\}$ over the generated $\sigma $-ring $\sigma (\mathbb \{A\})$: it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space $X$ over its Čech-Stone compactification $\beta X$ (or as the extension of continuous functions on $X$ over its Hewitt realcompactification $\upsilon X$). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that $\sigma (\mathbb \{A\})$ is the sequential envelope of $\mathbb \{A\}$ with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category $\mathop \{\{\mathrm \{I\}D\}\}$ of $-posets of fuzzy sets (such $-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on $\mathbb \{A\}$ over $\sigma (\mathbb \{A\})$ is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems.},
author = {Frič, Roman},
journal = {Mathematica Bohemica},
keywords = {extension of measure; categorical methods; sequential continuity; sequential envelope; field of subsets; D-poset of fuzzy sets; effect algebra; epireflection; categorical method; sequential continuity; sequential envelope},
language = {eng},
number = {4},
pages = {397-407},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Extension of measures: a categorical approach},
url = {http://eudml.org/doc/249578},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Frič, Roman
TI - Extension of measures: a categorical approach
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 4
SP - 397
EP - 407
AB - We present a categorical approach to the extension of probabilities, i.e. normed $\sigma $-additive measures. J. Novák showed that each bounded $\sigma $-additive measure on a ring of sets $\mathbb {A}$ is sequentially continuous and pointed out the topological aspects of the extension of such measures on $\mathbb {A}$ over the generated $\sigma $-ring $\sigma (\mathbb {A})$: it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space $X$ over its Čech-Stone compactification $\beta X$ (or as the extension of continuous functions on $X$ over its Hewitt realcompactification $\upsilon X$). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that $\sigma (\mathbb {A})$ is the sequential envelope of $\mathbb {A}$ with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category $\mathop {{\mathrm {I}D}}$ of $-posets of fuzzy sets (such $-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on $\mathbb {A}$ over $\sigma (\mathbb {A})$ is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems.
LA - eng
KW - extension of measure; categorical methods; sequential continuity; sequential envelope; field of subsets; D-poset of fuzzy sets; effect algebra; epireflection; categorical method; sequential continuity; sequential envelope
UR - http://eudml.org/doc/249578
ER -