{
  "id": "1203.5191",
  "version": "v1",
  "published": "2012-03-23T07:25:36.000Z",
  "updated": "2012-03-23T07:25:36.000Z",
  "title": "On the convergence of double integrals and a generalized version of Fubini's theorem on successive integration",
  "authors": [
    "Ferenc Moricz"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let the function $f: \\bar{\\R}^2_+ \\to \\C$ be such that $f\\in L^1_{\\loc} (\\bar{\\R}^2_+)$. We investigate the convergence behavior of the double integral $$\\int^A_0 \\int^B_0 f(u,v) du dv \\quad {\\rm as} \\quad A,B \\to \\infty,\\leqno(*)$$ where $A$ and $B$ tend to infinity independently of one another; while using two notions of convergence: that in Pringsheim's sense and that in the regular sense. Our main result is the following Theorem 3: If the double integral (*) converges in the regular sense, or briefly: converges regularly, then the finite limits $$\\lim_{y\\to \\infty} \\int^A_0 \\Big(\\int^y_0 f(u,v) dv\\Big) du =: I_1 (A)$$ and $$\\lim_{x\\to \\infty} \\int^B_0 \\Big(\\int^x_0 f(u,v) du) dv = : I_2 (B)$$ exist uniformly in $0<A, B <\\infty$, respectively; and $$\\lim_{A\\to \\infty} I_1(A) = \\lim_{B\\to \\infty} I_2 (B) = \\lim_{A, B \\to \\infty} \\int^A_0 \\int^B_0 f(u,v) du dv.$$ This can be considered as a generalized version of Fubini's theorem on successive integration when $f\\in L^1_{\\loc} (\\bar{\\R}^2_+)$, but $f\\not\\in L^1 (\\bar{\\R}^2_+)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-03-23T07:25:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "double integral",
      "fubinis theorem",
      "generalized version",
      "successive integration",
      "regular sense"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.5191M"
    }
  }
}