In the first volume of Feller's book (see at Archive), chapter 10 section 2, there is a proof of the weak law of large numbers for the case the second moment does not exist.
I am reading about 7 books in English and Russian and none goes further than using Chebyshev's Inequality and assuming existence of the first two moments. Wikipedia and just googling for a proof without variance do not help to find it either.
Does anyone know why such a beautiful and, moreover, much general proof is so unpopular? Or, maybe there is some imperfection I do not notice, could you point it out? Thank you!