Decision theory – the theory of decision making under risk – raises various questions about large infinities (infinite utilities) and about small infinities (infinitesimal probabilities). There are various puzzle cases – for example, the St. Petersburg Game and the Two-Envelope Paradox – which put pressure on the suggestion that we can allow infinite utilities into decision theory. There are other puzzle cases – for example, the case of an infinite fair lottery – which might be taken to suggest that we should be cautious about supposing that probability measures are more than finitely additive.

We shall begin with the mathematics of probability theory and with a discussion of additivity principles for probability measures. Next, we shall turn to decision theory and a discussion of infinite utility. With that discussion behind us, we shall move on to a discussion of some hard cases that bring out the potential costs of various assumptions about the infinitely large and the infinitely small in the theories of probability and decision.

PROBABILITIES

We begin with some mathematical preliminaries.

If ¥ is a nonempty set of subsets of a nonempty set X, then ¥ is an algebra of subsets of X iff ¥ is closed under finite unions and complementation. An algebra of subsets ¥ is a σ -algebra iff ¥ is closed under countable unions. If ℭ is a nonempty set of subsets of a nonempty set X, then the algebra (σ-algebra) generated by ℭ is the intersection of all algebras (σ-algebras) that include ℭ.