{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $I$ be a nonempty index set and, for each $i \\in I$, let $(X_i, \\Sigma_i)$ be a [[measurable space|N4KDT83V3S]].\nLet $X = \\prod_{i \\in I} X_i$ be the [[Cartesian product|09CEDJYBKP]], and let $\\pi_i \\colon X \\to X_i$ be the coordinate projections.\nThe *product σ-algebra* $\\bigotimes_{i \\in I} \\Sigma_i$ is the [[σ-algebra|0SV8K0TDGY]] on $X$ generated by the sets $\\pi_i^{-1}(A)$ with $i \\in I$ and $A \\in \\Sigma_i$.\nEquivalently, it is the smallest σ-algebra on $X$ for which every $\\pi_i$ is [[measurable|YYNCJSR0PJ]].\n\n## Remarks\n\nFor two factors, $\\Sigma_1 \\otimes \\Sigma_2$ is generated by the rectangles $A_1 \\times A_2$ with $A_1 \\in \\Sigma_1$ and $A_2 \\in \\Sigma_2$, since $A_1 \\times A_2 = \\pi_1^{-1}(A_1) \\cap \\pi_2^{-1}(A_2)$.\nFor general $I$, it is generated by the cylinder sets $\\prod_{i \\in J} A_i \\times \\prod_{i \\notin J} X_i$ with $J \\subseteq I$ finite.\n\nExample: $\\mathcal{B}(\\R) \\otimes \\mathcal{B}(\\R) = \\mathcal{B}(\\R^2)$, where $\\mathcal{B}$ denotes the [[Borel σ-algebra|0WQXY32Q8D]].\nEach rectangle of Borel sets is Borel in $\\R^2$, which gives one inclusion.\nEach open set of $\\R^2$ is a countable union of open rectangles with rational corners, which gives the other.\n\nNon-example: let $\\mathcal{L}$ be the σ-algebra of Lebesgue measurable subsets of $\\R$.\nThen $\\mathcal{L} \\otimes \\mathcal{L}$ is strictly smaller than the σ-algebra of Lebesgue measurable subsets of $\\R^2$.\nFor a non-measurable $N \\subseteq \\R$, the set $\\set{0} \\times N$ is a Lebesgue null set in $\\R^2$, but its section at $0$ is $N \\notin \\mathcal{L}$, and every set in $\\mathcal{L} \\otimes \\mathcal{L}$ has measurable sections.\n\nA function $f \\colon Z \\to X$ is measurable for $\\bigotimes_i \\Sigma_i$ if and only if every coordinate $\\pi_i \\circ f$ is measurable.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "6HVCTJGB61",
    "kind": "definition",
    "msc": [
      "28A05",
      "28A35"
    ],
    "name": "Product sigma-algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/%CE%A3-algebra"
      }
    ],
    "tag": "M8X6VZD91K",
    "uses": [
      "0SV8K0TDGY",
      "N4KDT83V3S",
      "09CEDJYBKP",
      "YYNCJSR0PJ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/M8X6VZD91K/",
  "links": [
    {
      "field": "defines",
      "target": "6HVCTJGB61"
    },
    {
      "field": "uses",
      "target": "0SV8K0TDGY"
    },
    {
      "field": "uses",
      "target": "N4KDT83V3S"
    },
    {
      "field": "uses",
      "target": "09CEDJYBKP"
    },
    {
      "field": "uses",
      "target": "YYNCJSR0PJ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/M8X6VZD91K.md",
  "msc": [
    "28A05",
    "28A35"
  ],
  "name": "Product sigma-algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M8X/M8X6VZD91K.md",
  "status": null,
  "tag": "M8X6VZD91K",
  "uri": "m2://M8X6VZD91K",
  "url": "https://math2.tamnd.com/wiki/product-sigma-algebra-definition/",
  "writing": {
    "tag": "M8X6VZD91K",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}