PreprintCC BY 4.0arXiv
Naturality characterizes product measures and relative product measures
Abstract
We show that the formation of the product measure is the only natural way to assign to each tuple of probability measures on measurable spaces a probability measure on the product space. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard Borel setting, we also prove an analogous result for relative products over a fixed base probability space. As a corollary, we conclude that the formation of relative products is a lax symmetric monoidal functor.
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BibTeX
@misc{bloch2026naturality,
title = {{Naturality characterizes product measures and relative product measures}},
author = {Victor Bloch and Tobias Fritz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15909},
eprint = {2609.15909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15909},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Naturality characterizes product measures and relative product measures
AU - Victor Bloch
AU - Tobias Fritz
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.15909
UR - https://arxiv.org/abs/2609.15909
AB - We show that the formation of the product measure is the only natural way to assign to each tuple of probability measures on measurable spaces a probability measure on the product space. Here, naturality is meant in the sense of category theory and amounts to the condition that the assignment commutes with pushforward along measurable maps. In the standard Borel setting, we also prove an analogous result for relative products over a fixed base probability space. As a corollary, we conclude that the formation of relative products is a lax symmetric monoidal functor.
ER -