Durrett 1.2

1.2

1.2.1

For any , . Then

1.2.2

lower bound:

upper bound:

1.2.3

For each set .

Set . Then, by the definition above,

A sum of uncountable positive values is infinite, so must be at most countable.

1.2.4

Set to be the distribution function of . Then for ,

Since maps into , for and for .

Set .

Then there must exist at least one s.t. .

Set .

1.2.5

From the definition of density, set

Then for with distribution function :

Next define the push-forward measure given by . Then using (i) change of variables and (ii) change of measures:

1.2.6

The density function of X is

Then the density function of is

1.2.7

Then differentiating:

Then for ,