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Unformatted text preview: 5 Integration We will first discuss the question of integrability of bounded functions on closed intervals, followed by the integrability of continuous functions (which are nicer), and then move on to bounded functions with negligible disconti nuities. The main tool will be to approximate the integral from above by the upper sum and from below by the lower sum , relative to various parti tions. This method was introduced by the famous nineteenth century Ger man mathematician Riemann, and it is customary to call these sums Rie mann sums . 5.1 Basic Notions Definition 5.1 If f is a bounded function on a closed interval [ a, b ] , then the span of f on [ a, b ] is given by span f ([ a, b ]) = sup f ([ a, b ]) − inf f ([ a, b ]) , where sup f ([ a, b ]) (resp. inf f ([ a, b ]) ) denotes the supremum (resp. infimum) of the values of f on [ a.b ] . If f is continuous on [ a, b ], then we know that it is bounded, and moreover, sup = max and inf = min (of f ([ a, b ])). Definition 5.2 A partition of a closed interval [ a, b ] is a collection of points t , t 1 , t 2 , . . . , t n such that t = a < t 1 < t 2 < ··· < t n − 1 < b = t n . Definition 5.3 A function S defined on [ a, b ] is called a step function if there is a partition P = { t , . . . , t n } of [ a, b ] , and constants c 1 , c 2 , . . . , c n such that such that S ( x ) = c j if x ∈ [ t j − 1 , t j ) , and S ( b ) = c n . A proper definition of integration must allow such a (step) function to be integrable, with its integral over [ a, b ], denoted ∫ b a S , being the sum ∑ n j =1 c j ( t j − t j − 1 ). 1 Definition 5.4 If P, P ′ are partitions of [ a, b ] , we will say that P ′ is a re finement of P iff the set of points in P is contained in the set of points of P ′ . For example, P : t = 0 < t 1 = 1 2 < t 2 == 1 and P ′ : t ′ = 0 < t ′ 1 = 1 4 < t ′ 2 = 1 2 < t ′ 3 = 3 4 < t ′ 4 = 1 are both partitions of [0 , 1], with P ′ a refinement of P . It is clear from the definition that given any two partitions P, P ′ of [ a, b ], we can find a third partition P ′′ which is simultaneously a refinement of P and of P ′ . Such a P ′′ is called a common refinement of P, P ′ . Remark : The sum and the product of two step functions are also seen to be step functions, by considering suitable refinements. Here is a quick definition of integrability: Definition 5.5 A bounded function f on [ a, b ] is integrable iff for every ε > , we can find a partition P = t = a < t 1 < t 2 < ··· < t n − 1 < b = t n such that the sum ∆ f ( P ) := n ∑ j =1 ( t j − t j − 1 )span f ([ t j − 1 , t j ]) is less than ε . Note that, for each ε > 0, the choice of P may depend on ε ....
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 Winter '08
 Borodin,A
 Order theory, dx, Monotonic function

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