Remark

— 1) The relation (11) does not necessarily hold when $\mathfrak{F}$ is an uncountable directed set of functions $\geq 0$ that are not lower semi-continuous. Take for example $X = \mathbf{R}$, $\mu$ being Lebesgue measure on $\mathbf{R}$, and consider the directed (for $\leq$) set $\mathfrak{F}$ of characteristic functions $\varphi_M$ of all the finite subsets M of $\mathbf{R}$. Then $\mu^*(\varphi_M) = 0$ for every finite subset M, because a point is contained in an open interval of arbitrarily small length, and the characteristic function of a set reduced to a point therefore has upper integral zero by Def. 3 and Prop. 9 of No. 2. But the upper envelope of $\mathscr{F}$ is the constant function equal to 1, and $\mu^*(1) = +\infty$.

2) Note that for a decreasing sequence $(f_n)$ of functions $\geqslant 0$, one does not necessarily have $\mu^*(\inf_n f_n) = \inf_n \mu^*(f_n)$, even if $\mu^*(f_n) < +\infty$ for all $n$ (cf. §4, Exer. 8 c)).