Prove that $Q$ is a probability measure

Let $(\Omega,\mathcal{A},\nu)$ be a measure space. Show that if the measure over the space $\Omega$ is $0<\nu(\Omega)<\infty$, then

$$Q=\frac{1}{\nu(\Omega)}\nu$$

is a probability measure.

I don't see how this is indeed a probability measure. For it to be a probability measure indeed it has to

• Return values in $[0,1]$.
• Satisfy countable additivity.
• Implied measure of $\emptyset$ should be $0$.

The only thing I currently know is that $\nu$ is a measure and that it is finite (and greater than $0$) over $\Omega$. This one seems quite tough.

Solution 1:

It is not tough. See that $Q(\Omega) = \frac{\nu(\Omega)}{\nu(\Omega)}=1$, and by monotonicity, your first criteria is satisfied. Countable additivity follows from the countable additivity of $\nu$ directly, and

$Q(\emptyset) = \frac{\nu(\emptyset)}{\nu(\Omega)}=\frac{0}{\nu(\Omega)}=0$.