Theoretical Statistics: Topics for a Core Course.
You can get a copy of the book at libgen.
What is in this book? ??.
Chapter 01: Probability and Measure
Measures:
A measure on a set assigns a non-negative value to many (but not necessarily all) subsets of , e.g., counting measure and Lebesgue measure. The theory of measure is fundamentally linked to basic questions about integration.
It is often impossible to assign measures to all subsets of . Instead, the domain of a measure will be a sigma field.
Definition: A collection of subsets of a set is a -field (or a -algebra) if: (a) , (b) If , then , (c) If , then .
Definition: A function on a -field of is a measure if: (a) For every , ; that is, . (b) If are pairwise disjoint elements of , then .
(Why ??) One useful consequence of the second part is that if are measurable sets with union called the limit of the sequence, then . This can be viewed as a continuity property of the measure.
Notation If is a -field of , the pair is called a measurable space and if is a measure on , the triple is called a measure space.
A measure is finite if and -finite if there exist sets in with and $$\big