I am learning Signed Measures as a Part of my PhD coursework and I have got this exercise after just learning 1 page.
I dont know how to proceed.
Let $\mu ,\nu $ ne two finite measures on the measurable space $(S,\sum)$.
Define $\mu \cup \nu =\mu+(\nu-\mu)^+$ and $\mu \cap \nu =\mu -(\mu-\nu)^+$
Prove that $\mu \cup \nu $ is the largest signed measure that is less than or equal to both $\mu,\nu$.
How to show that $\mu \cup \nu $ is the largest signed measure?What is meant by that?
Should I take a measurable set $M$ and show that $\mu \cup \nu(M)$ is greater than all other measures?
Any help?Any resources can help me.
The signed measures take values in $[-\infty,+\infty]$. That set has a linear order $\le$.
There then is a standard partial order on the signed measures on $(S,\Sigma)$:
$\mu_1 \le \mu_2$ iff $\forall A \in \Sigma: \mu_1(A) \le mu_2(A)$.
The largest signed measure that is $\le \mu, \nu$ is the measure $\lambda$ that obeys 2 properties:
$\lambda \le \mu$ and $\lambda \le \nu$ and for all other signed measures $\lambda'$ with $\lambda' \le \mu$ and $\lambda' \le \nu$ we have $\lambda' \ge \lambda$.
This is the standard definition of the infimum of two elements in a partial ordered set, and so I think you meant it to be for $\mu \cap \nu$, or more standard $\mu \land \nu$.
$\mu \cup \nu$ (really $\mu \lor \nu$) is the smallest signed measure $\ge$ both $\mu$ and $\nu$, the supremum.