STA3045F: Measurable Functions Summary

Chapter 3 MeasurableFunctions



Definition of a RV

Let r F IP be a probabilityspace

X r s IR be a function


X is a RV if X B E F HBE B IR


2 Definition of measurable



Let Cr F E be
E measure spaces



f
Let f r s E be a function



f is FIE measurable if f B e F H B E E
v

As CE E changes Thisjustmeans all preimages
we give the each in E mapback to sets
component special in F
names

, 1 If Cr F B IRPBCR


f

We f is borrel measurable
say
2 If Cr F IP is a probability space

Cr F B IR BCR
MF just means
f BCR 9 measurable

We call
f a random variable


If 4 is generated by blocks
3
Measurability on blocks

If F is generated by blocks
then function 4
any f r IR BCR
is BCIR A measurable



iff is constant on the blocks

Checking measurability for 4 not created by blocks

Let r F 31 E E be measurablesets