{
  "id": "1611.03630",
  "version": "v1",
  "published": "2016-11-11T09:16:44.000Z",
  "updated": "2016-11-11T09:16:44.000Z",
  "title": "A Category of Probability Spaces",
  "authors": [
    "Takanori Adachi",
    "Yoshihiro Ryu"
  ],
  "comment": "10 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We introduce a category Prob of probability spaces whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving direction of information with a way of its interpretation. We introduce a contravariant functor E from Prob to Set, the category of sets. The functor E provides conditional expectations along arrows in Prob, which are generalizations of the classical conditional expectations. For a Prob arrow f, we introduce two concepts f-measurability and f-independence and investigate their interaction with conditional expectations along f. We also show that the completion of probability spaces is naturally formulated as an endofunctor of Prob.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-11T09:16:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60A99",
      "16B50",
      "60G20",
      "18B99"
    ],
    "keywords": [
      "probability spaces",
      "category prob",
      "contravariant functor",
      "classical conditional expectations",
      "prob arrow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}