MEASURE AND INTEGRATION: LECTURE 1
Preliminaries. We need to know how to measure the size or vol
ume of subsets of a space X before we can integrate functions f : X
R or f : X C.
Were familiar with volume in Rn . What about more general spaces
X ? We nee
MEASURE AND INTEGRATION: LECTURE 2
Proposition 0.1. Let M be a algebra on X , let Y be a topological space, and let f : X Y . (a) Let be a collection of sets E Y such that f 1 (E ) M. Then is a algebra on Y . (b) If f is measurable and E Y is Borel, then
MEASURE AND INTEGRATION: LECTURE 3
Riemann integral. If s is simple and measurable then
N
sd =
i (Ei ),
X
i=1
N
where s = i=1 i Ei . If f 0, then
f d = sup
sd | 0 s f , s simple & measurable .
X
X
Recall the Riemann integral of function f on interval [a,
MEASURE AND INTEGRATION: LECTURE 4
Integral is additive for simple functions.
Proposition 0.1. Let s and t be nonnegative measurable simple func
tions. Then X (s + t)d = X s d + X t d.
Proof. Let E M and dene (E ) = E s d. First we show that is
measurable
MEASURE AND INTEGRATION: LECTURE 5
Denition of L1 . Let f : X [, ] be measurable. We say that f is n L1 (written f L1 () or simply f L1 ) X f + d < i and X f d X |f | d < . Dene + f d = f d f d
X X X
when at least one of the terms on the righthand side