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
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
