additivity - THE ADDITIVITY OF THE INTEGRAL FOKKO VAN DE...

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THE ADDITIVITY OF THE INTEGRAL FOKKO VAN DE BULT In this note we prove the following Theorem. Theorem 1. Let f and g be integrable functions on the interval [ a,b ] . Then f + g is also integrable on [ a,b ] and we have Z b a ( f + g )( x ) dx = Z b a f ( x ) dx + Z b a g ( x ) dx. Proof. The proof given here is almost the same as the one described in the book on pages 85 and 86, and probably written down better there. Suppose f and g are integrable on [ a,b ]. Choose an arbitrary n N . Then we see that there exists a step function s n f such that R b a s n ( x ) dx I ( f ) - 1 4 n = R b a f ( x ) dx - 1 4 n , as otherwise I ( f ) - 1 4 n would be an upperbound to S = { R b a s ( x ) dx | s f,s a step function } less than I ( f ), in violation of the definition I ( f ) = sup( S ). Likewise we find t n , ˜ s n and ˜ t n with R b a t n ( x ) dx R b a f ( x ) dx + 1 4 n , R b a ˜ s n ( x ) dx R b a g ( x ) dx - 1 4 n and R b a ˜ t n ( x ) dx R b a g ( x ) dx + 1
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additivity - THE ADDITIVITY OF THE INTEGRAL FOKKO VAN DE...

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