Fourth axiom in Kolmogorov's probability axiom system?

Question

I'm reading Jaynes's "Probability Theory: The Logic of Science". In appendix A he contrasts the theory he develops in the first few chapters to the more conventional measure theory based system, A.K.A. Kolmogorov's. He lists four axioms used in this system. $F$ is a $\sigma$-algebra (He calls it a "$\sigma$-field")

(1) Normalization: $P(\Omega) = 1$

(2) Non-negativity: $P(f_{i}) ≥ 0$ for all $f_{i}$ in $F$

(3) Additivity: if ${f_{1}\ldots f_{n}}$ are disjoint elements of $F$ then $P( f ) = \sum_{i}P(f_{i})$, where $f =\cup_{j}f_{j}$

(4) Continuity at zero: if a sequence $f_{1} \supseteq f_{2} \supseteq f_{3} \supseteq\ldots $ tends to the empty set, then $P( f_{j} ) → 0$.

I remembered use of only the first three axioms, was the fourth once used and abandoned or is there something else going on?

Answer 1

There is an alternate set of axioms that appear in my book (An Introduction to Mathematical Statistics and Its Applications by Larsen and Marx) which more clearly explain what types of functions can qualify as probability measures. They replace your fourth axiom by the following:

4. Let $A_1$, $A_2$, $\cdots$, be events defined over $\Omega$. If $A_i \cap A_j = \emptyset$ for each $i \neq j$, then $$ P\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} P(A_i) $$

In conjunction with the third axiom, this one is trying to say that the probability measure chosen must work for countably infinite disjoint decompositions of $\Omega$ as well. The third axiom does not automatically guarentee this unless the chosen probability measure converges to zero as the sequence of $f_i$ tends to the empty set. In short, the fourth axiom from your list or from my list is required to allow infinite decompositions of the sample set.

See also this answer.

Edit: In response to your comment, see also this answer to see some examples of measures that are finitely additive but not countably additive.