Daniell integral

Remarks

Example: let $H$ be the continuous functions $\R \to \R$ with compact support, and $I(h) = \int h(x) \, dx$, the Riemann integral. Linearity and positivity are clear. For clause 3, all $h_n$ vanish outside a compact interval $K$ that contains the support of $h_1$. By Dini's theorem $h_n \to 0$ uniformly on $K$, so $0 \le I(h_n) \le \max h_n \cdot \operatorname{length}(K) \to 0$.

Non-example: let $H$ be the convergent real sequences on $X = \N$, and $I(h) = \lim_{n \to \infty} h(n)$. Clauses 1 and 2 hold. Clause 3 fails: $h_n = \mathbf{1}_{\set{n, n+1, \dots}}$ decreases to $0$ pointwise, but $I(h_n) = 1$ for all $n$.

Daniell's construction extends $I$ to a class of integrable functions on which the monotone and dominated convergence theorems hold.