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fundamental theorems of calculus for Lebesgue integration

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Loosely, the Fundamental Theorems of Calculus serve to demonstrate that integration and differentiation are inverse processes. Suppose that is an absolutely continuous function on an interval $[a,b]\subset\mathbb{R}$ . The two following forms of the theorem are equivalent.

First form of the Fundamental Theorem:

There exists a function Lebesgue-integrable on such that for any $x\in [a,b]$ , we have $F(x)-F(a)=\int_a^x f(t) dt$ .

Second form of the Fundamental Theorem:

is differentiable almost everywhere on and its derivative, denoted , is Lebesgue-integrable on that interval. In addition, we have the relation $F(x)-F(a)=\int_a^x F'(t)dt$ for any $x\in [a,b]$ .

"fundamental theorems of calculus for Lebesgue integration" is owned by mathcam. [ full author list (3) | owner history (4) ]

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Other names:

first fundamental theorem of calculus, second fundamental theorem of calculus, fundamental theorem of calculus

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26-00 (Real functions :: General reference works )

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