1

Problem

Consider the following set:

For measures $\mu , \nu$ on the measurable space $(\Omega, \mathscr{F})$ where $\mu (\Omega) = \nu (\Omega) < + \infty$, consider the set $D = ${$ A \in \mathscr{F} : \mu (A) = \nu(A)$ }. We want to determine if this is a $\sigma$-algebra.

This is a set that is often used as a standard example of a d-system (Dynkin system). I have found that this is not a $\sigma$ algebra, however, it is unclear to me how to construct a counterexample to show this fact.

Progress

Of course, given the fact that it is a d-system means that it will definitely contain $\Omega$ and will be closed under complements. Therefore, in order to find a counterexample, we must look for measures that ensure that the set $D$ is not closed under countable unions.

Looking online has produced a few different examples of possible counterexamples, however, as this was an exercise in a textbook I am working through, I am more interested in understanding how to construct a counterexample as opposed to just copying down a specific example that works in this instance without actually learning to construct this on my own.

Example Counterexample

The counterexample used in the textbook is the following:

Consider $D$ (as defined above) in the following setup:

Sample space: $\Omega = ${$1,2,3,4$}; Sigma Algebra: $\mathscr{F} = P(\Omega)$ (power set); Measures: $\mu = δ_{\{1\}} +δ_{\{2\}}$ and $\nu = δ_{\{3\}} +δ_{\{4\}}$.

If I didn't have access to the solutions, I never would have been able to construct this example on my own.

What is the intuition behind something like this (or behind any possible answer for this question)? How can I learn to construct examples of my own here?

FD_bfa
  • 3,989
  • 1
    To find this or any other example, it can be more manageable to make the property of "being closed with respect to finite intersections" fail. You can forget about the measures first and just produce a Dynkin system without this property. Then define the measures later observing that they must be equal at the sets of the system. – plop Jan 05 '23 at 04:09
  • Yes, it is not too important. They chose the smallest one to make it pretty. – plop Jan 05 '23 at 04:15
  • I haven't tried myself. Can you produce a Dinkin system that is not a $\sigma$-algebra on that set? – plop Jan 05 '23 at 04:19
  • I don't believe it's possible, but if we go for a slightly different example such as the one defined in the top answer here: https://math.stackexchange.com/questions/8173/what-is-an-example-of-a-lambda-system-that-is-not-a-sigma-algebra how do we now define measures to make this particular example work? – FD_bfa Jan 05 '23 at 04:34

1 Answers1

1

My way of thinking about it after reading the first lines of your question.

It is enough to find a situation with $\mu(A)=\nu(A)$, $\mu(B)=\nu(B)$ and $\mu(A\cup B)\neq\nu(A\cup B)$.

A handsome partition of $\Omega$ that comes to mind is then the collection: $$\mathcal P=\{A\cap B,A\cap B^c,A^c\cap B,A^c\cap B^c\}$$ We need a $\sigma$-algebra that contains the elements of $\mathcal P$ and keeping things small and simple we take the collection of unions of elements of $\mathcal P$. Every measure on this $\sigma$-algebra is completely determined by the values that it takes on the elements of $\mathcal P$.

Now is it possible to find values for $\mu$ and $\nu$ on these sets in such a way that the conditions mentioned above are satisfied?

Yes, it is.

As an example let $\mu(P)=2$ for every $P\in\mathcal P$ and let $\nu(A\cap B)=\nu(A^c\cap B^c)=1$ and $\nu(A\cap B^c)=\nu(A^c\cap B)=3$.

Then $\mu(A)=\nu(A)=4$, $\mu(B)=\nu(B)=4$ and $\mu(A\cup B)=6\neq7=\nu(A\cup B)$.

Also we have $\mu(\Omega)=\nu(\Omega)=8<+\infty$.

The Dynkin-system that arises from this is:$$\mathcal D=\{\varnothing, A, B, A^c,B^c,\Omega\}$$ where the sets in $\mathcal P=\{A\cap B,A\cap B^c,A^c\cap B,A^c\cap B^c\}$ are not empty.

drhab
  • 151,093