LECTURE NOTES IN MEASURE THEORY - Chalmers

1 LECTURE NOTESIN MEASURE THEORYC hrister BorellMatematikChalmers och G teborgs universitet412 96 G teborg(Version: January 12)2 PREFACET hese are LECTURE NOTES on integration THEORY for a eight-week course at theChalmers University of Technology and the G teborg University. The partsde ning the course essentially lead to the same results as the rst threechapters in the Folland book[F];which is used as a text book on the proofs in the LECTURE NOTES sometimes di er from those given in[F]:Hereis a brief description of the di erences to simplify for the Chapter 1 we introduce so called -systems and -additive classes,which are substitutes for monotone classes of sets[F]. Besides we prefer toemphasize metric outer measures instead of so called premeasures. Through-out the course, a variety of important measures are obtained as image mea-sures of the linear MEASURE on the real line.

If fis a function from a set Ainto a set B;this means that to every x2 A there corresponds a point f(x) 2 Band we write f: A! B:A function is ... is a measurable space and the function : M ! [0;1] is a positive measure, (X;M; ) is called a positive measure space. The quantity (A) is called the -measure of Aor simply the measure of Aif there is

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