(a) In what way are the theorems of Fubini and Clairaut similar? (b) If f(x, y) is continuous on [a, b] ×[c, d] and g(x, y)=∫a^x ∫c^y f(s, t) d t d s for a<x<b, c<y<d, show that gx y=gy x=f(x, y) | Numerade
(a) In what way are the theorems of Fubini and Clairaut similar? (b) If f…
May 23, 2021
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Julia | Math & Programming on Instagram: "Flip them around, Fubini-Tonelli: Let X, µ and Y, ν be measure spaces with corresponding σ-algebras A, B. Assume X and Y are σ-finite. Theorem: (Fubini-Tonelli) For complex-valued measurable f, g, if any one of ∫_X ∫_Y |f(x, y)| dµ(x) dν(y) ∫_Y ∫_X |f(x, y)| dν(y) dµ(x) ∫_X×Y |f(x, y)| dµ×ν is finite, then they all are finite, and are equal. For [0, +∞]-valued functions f, ∫_X ∫_Y f(x, y) dµ(x) dν(y) = ∫_Y ∫_X f(x, y) dν(y) dµ(x) = ∫_X×Y f(x, y) dµ×ν alt
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