View source for Fubini's theorem on differentiation

In mathematics, '''Fubini's theorem on differentiation''', named after [[Biography:Guido Fubini|Guido Fubini]], is a result in [[Real analysis|real analysis]] concerning the [[Derivative|differentiation]] of series of [[Monotonic function|monotonic function]]s. It can be proven by using [[Fatou's lemma]] and the properties of [[Null set|null set]]s.<ref name="a">Jones, Frank (2001), ''Lebesgue Integration on Euclidean Space'', Jones and Bartlett publishers, pp.&nbsp;527–529.</ref>
== Statement ==
Assume <math>I \subseteq \mathbb R</math> is an [[Interval (mathematics)|interval]] and that for every [[Natural number|natural number]] ''k'', <math>f_k: I \to \mathbb R</math> is an increasing function. If, 

:<math>s(x) := \sum_{k=1}^\infty f_k(x)</math>

exists for all <math>x \in I,</math> then for [[Almost everywhere|almost any]] <math>x \in I,</math> the derivatives exist and are related as:<ref name="a"/> 

:<math>s'(x) = \sum_{k=1}^\infty f_k'(x).</math>

In general, if we don't suppose ''f<sub>k</sub>'' is increasing for every ''k'', in order to get the same conclusion, we need a stricter condition like [[Uniform convergence|uniform convergence]] of <math>\sum_{k=1}^n f_k'(x)</math> on&nbsp;''I'' for every n.<ref>Rudin, Walter (1976), ''Principles of Mathematical Analysis'', McGraw-Hill, p.&nbsp;152.</ref>

== References ==
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[[Category:Theorems in real analysis]]
[[Category:Theorems in measure theory]]

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