Corollary 5

— Let $\mu$ and $\nu$ be two positive measures on T. The conditions 1), 2), 3) of Th. 2 are also equivalent to the following ones:

4) For every $\nu$-integrable numerical function $f \geq 0$ and for every number $\varepsilon > 0$, there exists a $\delta > 0$ such that the relations $0 \leq h \leq f$ and $\int^* h \, d\mu \leq \delta$ imply $\int^* h \, d\nu < \varepsilon$.

5) For every function $g \in \mathscr{K}_+(\mathrm{T})$ and every number $\varepsilon > 0$, there exists a $\delta > 0$ such that, for every $h \in \mathscr{K}_+(\mathrm{T})$ bounded above by $g$ and satisfying $\int h \, d\mu \leq \delta$, one has $\int h \, d\nu \leq \varepsilon$.

6) For every compact set $K \subset \mathrm{T}$ and every number $\varepsilon > 0$, there exists a $\delta > 0$ such that the relations $A \subset K$ and $\mu^*(A) \leq \delta$ imply $\nu^*(A) \leq \varepsilon$.

The implications 4) $\Rightarrow$ 6) $\Rightarrow$ 3) are obvious.

Suppose there exists a finite function $k \geq 0$, universally measurable and locally $\mu$-integrable, such that $\nu = k \cdot \mu$, and let us show that the condition 4) is satisfied. Let $f$ be a $\nu$-integrable function $\geq 0$, and let $\varepsilon > 0$. For every integer $n \geq 0$, let $A_n$ be the set of $t \in \mathrm{T}$ such that $k(t) \geq n$. The functions $f \varphi_{A_n}$ form a decreasing sequence, bounded above by $f$ and tending pointwise to 0, therefore there exists an $N$ such that $\int f \varphi_{A_N} \, d\nu \leq \varepsilon/2$ (Ch. IV, §4, No. 3, Prop. 4). If $h$ is a function on $\mathrm{T}$ satisfying $0 \leq h \leq f$ and $\int^* h \, d\mu \leq \varepsilon/2N$, then

We have thus proved that the conditions 4) and 6) are equivalent to the conditions of Th. 2.

It is clear that 4) implies 5). Finally, if the condition 5) is satisfied, then $\nu$ belongs to the band generated by $\mu$ (Ch. II, §2, No. 2, Prop. 5), hence has base $\mu$ (Cor. 2).