Proposition 3

a) The set $\mathscr{L}_u(X)$ is an involutive subalgebra of the involutive algebra of functions from X into $\mathbf{C}$;

b) The set $\mathscr{L}_u^{\infty}(X)$ is a subalgebra of $\mathscr{L}_u(X)$ and the mapping defined by $f\mapsto \|f\|_{\infty}$ is a norm on $\mathscr{L}_u^{\infty}(X)$ for which $\mathscr{L}_u^{\infty}(X)$ is a unital stellar algebra;

c) The algebra $\mathscr{L}_u(X)$ contains $\mathscr{C}(X)$ and $\mathscr{L}_u^{\infty}(X)$ contains $\mathscr{C}_b(X)$;

d) Every Borel function from X into $\mathbf{C}$ (TG, IX, p. 61, def. 5) belongs to $\mathscr{L}_u(X)$;

e) For every sequence $(f_n)_{n\in\mathbf{N}}$ in $\mathscr{L}_u(X)$ which converges simply to a function $f$ from X into $\mathbf{C}$, one has $f\in \mathscr{L}_u(X)$.

Assertions a) and c) follow from the definitions (cf. INT, IV, p. 175, § 5, No$^o3$, cor. 3). Assertion d) is a consequence of INT, IV, p. 179, § 5, No$^o5$, th. 4, since the inverse image under a Borel function of every Borel subset of $\mathbf{C}$ is a Borel subset of X, which is universally measurable (INT, V, p. 28, § 3, No$^o4$). Finally, assertion e) follows from Egoroff's theorem (INT, IV, p. 175, § 5, No$^o4$, th. 2).

To prove assertion b), it is enough to remark that e) implies, a fortiori, that a uniform limit of universally measurable functions is universally measurable.

There may exist universally measurable functions on X which are not Borel. In fact, if X is metrisable, the characteristic function of a Souslin subset of X is universally measurable (cf. INT, IV, p. 171, § 5, No$^o1$, cor. 2); now, in every uncountable Polish space, there exists a Souslin subset which is not Borel ("Souslin's theorem"; this follows from TG, IX, p. 120, exercise 8 and from the fact that for every uncountable Polish space X, there exists a Borel bijection $X\rightarrow \mathbf{N}^{\mathbf{N}}$ whose inverse is Borel).