$\textit{Definition:}$ The information structure $\mathfrak{I}=((X^i),\mu)$ is given by a finite set of signals $X^i$ for each $i$ and by a probability measure $\mu$ over $X$. When $x$ is drawn according to$\mu$, player $i$ is informed about the coordinate $x^i$.
$\textit{Question:}$ Why do we need to define the information structure as a measure set which is finite and what does it mean that we know the probability measure?
I would appreciate it if anybody could give some example as well.
It's a finite set, so it's measurable, that's for free. In fact I would wager that it is chosen to be finite merely to do away with measure theoretic technicalities. The question you're actually asking, it seems to me, is why there's a probability distribution involved at all, that is, why is $X$ random. This ultimately has to get down to what is being modeled. As per the paper you linked to in your other questions, this structure was proposed by Aumann in an earlier paper. Did you read that? That's perhaps the cleanest way to get something definitive. One can make some general observations though.
Notice that the basic structure of this situation is that there is some central mechanism that communicates information to each player, who can then adapt their strategies on the basis of this information. Keep in mind that this central mechanism is a useful abstraction, and should not be taken too literally.
One situation we may model in this way is if all the players are probing some common environment - so 'the world' has some state, and the players make observations of this state to adapt their play. Most situations of this sort are noisy - typically the underlying state is random, and our measurement processes introduce further noise into the observation. The definition handles this by taking all this noise, and neatly packaging it into one distribution $\mu$, which is ultimately all that affects the agents' play (since they can only act on the basis of their $X^i$s).
As a broader comment, I think you might be skimping on your reading. At a basic level, the way theoretical study works is that you start with broad observations of the subject you care about, model these to abstract out the messy details (and to simplify enough to enable analysis), and then analyse this model to derive consequences that let you reason about how the actual thing might work (there's a lot of feedback that I'm ignoring here). I think there's a tendency for people starting out to think that all the action is in the third phase, but this is not true - in fact the first two phases are crucial, and also in a sense more difficult because they are softer. This question, along with the others you asked, suggest that you're not doing enough basic reading to get the overall structures the subject cares about, and are jumping ahead to ask why the particulars of the definitions (the second phase) are set up the way they are. Ultimately the reason for this is observations made in the first phase (and some amount of technical convenience), and so such things will remain gobbledygook until you grok something about the structures.
Do note that this is not easy, and takes time. Also in my opinion this is done much better verbally (in meetings/reading groups/class) than textually. Try to speak with your peers or an advisor for both references on what to read, and to discuss and clarify what you read in them.