Exercise 10, § 3

Let $X$ be an uncountable infinite set, equipped with a well-ordering relation denoted $x \leq y$. For every $x \in X$, one denotes by $I(x)$ the set of $y \in X$ such that $y < x$. We make the following hypotheses: $\alpha$ there exists a largest element $a$ in $X$; $\beta$ the set of cardinals strictly smaller than $\mathrm{Card}(X)$ has a largest element $c$; $\gamma$ for every $x$ in $I(a)$, one has $\mathrm{Card}(I(x)) \leq c$, and the set $Y$ of $x \in X$ such that $\mathrm{Card}(I(x)) = c$ is nonempty;

δ) every gauge on a set with cardinal $\leq c$ is atomic. Show that every gauge on X is atomic. One can argue by contradiction in the following way: let $m$ be a nonzero diffuse gauge on X. Denote by b the smallest element of Y, and for every $x \in I(a)$ let $f_x$ be an injection of $I(x)$ into $I(b)$. For every pair $(x, y) \in I(a) \times I(b)$, denote by $A_{x,y}$ the set of elements $z \in X$ such that $x < z < a$ and $f_z(x) = y$; for every $x \in I(a)$, denote by $M_x$ the set of $y \in I(b)$ such that $m(A_{x,y}) > 0$, and for every $y \in I(b)$, denote by $N_y$ the set of $x \in I(a)$ such that $m(A_{x,y}) > 0$. Using Exer. 9 c), show that $m(X) = \sum_{y \in I(b)} m(A_{x,y})$; from this, deduce that the set $M_x$ is countable and nonempty for every $x \in I(a)$; then show that each of the sets $N_y$ is countable and from this deduce the contradiction $\mathrm{Card}(X) = c$.