Is independence indpendent on measure?

[tunaaa]

Is it possible that 2 sigma-algebras could be independent under one measure but not independent under another?

Many thanks.

 

[chiro]

Is it possible that 2 sigma-algebras could be independent under one measure but not independent under another?

Many thanks.

Hey tunaaa and welcome to the forums.

I don't know much about measure theory so maybe you could give us the definition of independence for a measure. I've heard about decomposing measures into orthogonal parts but I don't think this is what you are asking about.

 

[micromass]

Hey tunaaa and welcome to the forums.

I don't know much about measure theory so maybe you could give us the definition of independence for a measure. I've heard about decomposing measures into orthogonal parts but I don't think this is what you are asking about.

He probably means independence wrt a probability measure. This is defined as follows: take a probability space $(\Omega,\mathcal{F},P)$ and take $\mathcal{B}_1$ and $\mathcal{B}_2$ sigma-algebra's which are part of $\mathcal{F}$. They are independent if for every $B_1\in \mathcal{B}_1$ and $B_2\in \mathcal{B}_2$ holds that

$$P(B_1\cap B_2)=P(B_1)P(B_2)$$

Like the definition suggests, the measure P is critical here. If we have another measure on $(\Omega,\mathcal{F}$, then independence must not hold.

For example, look at $(\Omega,\mathcal{F})=(\mathbb{N},\mathcal{P} (\mathbb{N}))$ and take $P_1$ uniquely defined by $P_1(\{0\})=1$. Further, take $P_2$ uniquely define by $P_2(\{0\})=P_2(\{1\})=1/2$.

Then {0} and {1} (which generate sigma-algebras) are independent for $P_1$, but dependent for $P_2$.

 

[tunaaa]

Many thanks

 

[blue_raver22]

I can say that independence is a mathematical concept that is independent of the measure being used. In other words, the independence of two events or sets is not affected by the choice of measure. Therefore, it is possible for two sigma-algebras to be independent under one measure but not under another. This is because the independence of two sigma-algebras is determined by their structure and not the measure being used.