Fubini's theorem on differentiation (original) (raw)

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In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets.[1]

Assume I ⊆ R {\displaystyle I\subseteq \mathbb {R} } $I\subseteq \mathbb {R}$ is an interval and that for every natural number k, f k : I → R {\displaystyle f_{k}:I\to \mathbb {R} } $f_{k}:I\to \mathbb {R}$ is an increasing function. If,

s ( x ) := ∑ k = 1 ∞ f k ( x ) {\displaystyle s(x):=\sum _{k=1}^{\infty }f_{k}(x)} $s(x):=\sum _{k=1}^{\infty }f_{k}(x)$

exists for all x ∈ I , {\displaystyle x\in I,} $x\in I,$ then for almost any x ∈ I , {\displaystyle x\in I,} $x\in I,$ the derivatives exist and are related as:[1]

s ′ ( x ) = ∑ k = 1 ∞ f k ′ ( x ) . {\displaystyle s'(x)=\sum _{k=1}^{\infty }f_{k}'(x).} $s'(x)=\sum _{k=1}^{\infty }f_{k}'(x).$

In general, if we don't suppose fk is increasing for every k, in order to get the same conclusion, we need a stricter condition like uniform convergence of ∑ k = 1 n f k ′ ( x ) {\displaystyle \sum _{k=1}^{n}f_{k}'(x)} $\sum _{k=1}^{n}f_{k}'(x)$ on I for every n.[2]

^ a b Jones, Frank (2001), Lebesgue Integration on Euclidean Space, Jones and Bartlett publishers, pp. 527–529.
^ Rudin, Walter (1976), Principles of Mathematical Analysis, McGraw-Hill, p. 152.