In the paper P. Erdős and A. Hajnal, On the structure of set-mappings, Acta Math. Acad. Sci. Hungar.9 (1958), 111-131, the authors came close to solving Ulam's measure problem by proving that the first uncountable inaccessible cardinal is not measurable, as they pointed out in their subsequent paper P. Erdős and A. Hajnal, Some remarks concerning our paper "On the structure of set-mappings" — non-existence of a two-valued $\sigma$-measure for the first uncountable inaccessible cardinal, Acta Math. Hungar.13 (1962), 223-226:
In accordance with the notations of [4] we say that a cardinal $m$ possesses property $P_3$ if every two-valued measure $\mu(X)$ defined on all subsets of a set $S$ of power $m$ vanishes identically, provided $\mu(\{x\})=0$ for every $x\in S$ and $\mu(X)$ is $m$-additive.
It was well known that $\aleph_0$ fails to possess property $P_3$ and that every cardinal $m\lt t_1$ possesses property $P_3$ where $t_1$ denotes the first uncountable inaccessible cardinal.
Recently A. Tarski has proved, using a result of P. Hanf, that a certain wide class of strongly inaccessible cardinals possesses property $P_3$ (called strongly incompact cardinals). H. J. Keisler gave a purely set-theoretical proof of this result. After having seen these papers we observed that the special case of this result that $t_1$ possesses property $P_3$ follows almost trivially from some of our theorems proved in [1]. We are going to give this simple proof in §2. Our method for the proof is of purely combinatorial character, and although it is certainly weaker than that of A. Tarski and H. J. Keisler, we think that it is of interest to formulate how far one can go with these methods at present.
