“One can get into great philosophical debates on what is randomness. Information that we can’t compress. Information that’s unpredictable. Information that we are willing to bet on. ”

Because of the existence of  uncertainty, we have to do some work to make ourselves sure about something to some extent.  So we have to involve the measure structure to analyze what we are concerned.  I think the analysis of the difference between the function and random variable is a good way to get the idea why we want to involve the randomness.

The random variable is nothing but a function with some measure structure sitting behind. For example, there is a function on the real line: y=f(x)=x. If we have the measure structure on the real line, for example, point mass on 2, then y could only take 2 almost surely. That is to say, the uncertainty on x leads to the uncertainty for the prediction for y. However, people always want to find out something which are with 100% or 99% certainty among the uncertain information. And you probably know the 3 sigma rule in the normal model, that is the variance could help a lot to make you more sure about something. And now what do you think of the SLLN and WLLN?