How many probability measure are there? I'm new to Probability Theory and I have learned that for a function $\mu$ to be a probability measure, it must satisfy a set of requirements.
I wonder whether for the same measurable space ($i.e.$ when we looking at the same thing/event), can we have 2 different probability measures and why ?
EDIT:
If we can have 2 different probability measures when "measure the same thing", is it counter-intuitive ? (like we have 2 different probability for the same thing to happen)
 A: Certainly we can have many measures on a measurable space. For example, the counting measure assigns measure $n$ to any event of size $n$, and infinite measure otherwise (though that's not a probability measure). The "point mass at $x$" measure, for any $x \in \Omega$, is also a measure - it's even a probability measure, for each $x$. In that case, you're saying that an event only has any chance to happen if it contains one specific outcome, though you're varying which exact outcome it is.
The measure essentially determines the probability distribution. If you change the measure, that means you're redistributing the probability mass between events - for example, maybe you're biasing the dice differently.
There's a useful theorem which tells you when two measures are the same. If they assign finite measure to $\Omega$ (which probability measures always do), and they agree on a $\pi$-system (that is, a collection of sets closed under intersection) generating $\mathcal{F}$, then they agree everywhere. But otherwise you can have a huge variety of measures on the measurable space.
A: Generally there are uncountably many possible ways to assign probabilities within the axioms of probability. For example in tossing a coin, you can assign to the outcome heads any real number $p\in[0,1]$. Then the outcome tails must be assigned the probability $1-p$ (unless you want to allow for some other outcomes such as the coin lands on its edge; or the coin rolls away and disappears and can't be found.)
However, in practical situations, the preferred probability distribution (or measure) is one that reflects (our view of) reality. We think the coin is fair, so heads and tails are both assigned a probability of $0.5$.  We certainly can assign a fair coin a different probability distribution; but then our probability distribution does not reflect reality, and will not be as useful.
