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For an algebra $A\subseteq C(K)$ on a compact space, its uniform closure consists of functions that are uniform limits of sequences from $A$. It is a closed algebra in the uniform norm, because addition and multiplication preserve uniform convergence for bounded functions. If it contains every constant and separates points, the Stone-Weierstrass theorem makes the closure all of $C(K)$.

A continuous self-map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure. An irrational rotation of the circle is uniquely ergodic: invariance multiplies its nonzero-index Fourier coefficients by nontrivial phases, forcing those coefficients to vanish. The Stone-Weierstrass theorem then identifies the invariant measure as normalized Lebesgue measure.