Here, we will give an aside on probabilities, and in particular on various ways to establish what is know as concentration. Given some information about a distribution, e.g., its mean or its variance or information about higher moments, there are various ways to establish bounds on the tails of sums of random variables from that distribution. That is, there are various ways to establish that es...

We give an extension of de Finetti’s concept of coherence to unbounded (but real-valued) random variables that allows for gambling in the presence of infinite previsions. We present a finitely additive extension of the Daniell integral to unbounded random variables that we believe has advantages over Lebesgue-style integrals in the finitely additive setting. We also give a general version of th...