measure

For a set X with -algebra on X. A measure on is a function such that:

2. for disjoint sets

Properties

Preserves Order

For sets

Proof

Since , such that or Because we know that

Measure of set difference

For sets

Proof

This one is pretty obvious…

Countable subadditivity

For measure space , and ,

Note: This is a big one. Very important theorem.

Proof

Key idea: Make the sets disjoint. Then we can use the definition of them measure. We want to construct sets that we can ‘subtract’ out of the other sets to make the sequence disjoint. Let and Then since each only takes out the part that has already been included in the union. Thus, since

Continuity of measure

1. Increasing union version: For sequence of sets (an increasing sequence of sets)

2. Decreasing intersection version: For sequence of sets (an decreasing sequence of sets) and

Proof

1. Since each set is a superset of the previous, we can ‘make’ them all disjoint. WLOG, let so the Then which is a disjoint union.

Thus,

2. This one depends on De Morgan’s law and then uses 1.