Theorem 2

Suppose that $u_N$ is of finite trace. Then the function $x\mapsto N(x, x)$ belongs to $\mathscr{L}^1(X)$ and we have

Tr($u_N$) $=\int_XN(x, x)d\mu(x)$.

From conditions (i) and (ii), we have $|N(x, x)|\leqslant H(x, x)$ for $x\in \widetilde{X}$, therefore the function $x\mapsto N(x, x)$ belongs to $\mathscr{L}^1(X)$ by condition (iii).

Condition (i) establishes the equality

for all $x\in \widetilde{X}$. From conditions (ii) and (iii), we may apply Lebesgue's theorem (INT, IV, p. 137, § 3, n$^o7$, th. 6), from which it follows that

$=\sum_{i\in I}\alpha_i\langle f_i|g_i\rangle =$ Tr($u_N$),

the last equality resulting from the cor. of prop. 17 of IV, p. 170.