Independence of two events, random variables, sigma-fields

One thing I forgot to mention in the video: it is usually more demanding to check that two random variables are independent than to check that they are not:

- To prove that two random variables are not independent, it suffices indeed to find *one* counter-example, namely two (Borel) sets B_1, B_2 such that P({X_1 in B_1, X_2 in B_2}) is not equal to P({X_1 in B_1}) P({ X_2 in B_2}).

- On the contrary, independence requires that equality holds for *all* (Borel) sets B_1, B_2. In that sense, it is much more demanding than asking that the two random variables are just uncorrelated (which boils down to checking that a single number Cov(X_1,X_2)=0, as we shall see next week).