Exercise 27, § 5

Let X be a locally compact space, μ a real measure on X, K a compact subset of X, (f_n) a sequence of bounded μ-measurable numerical functions defined on K and separating the points of K. Show that for every bounded μ-measurable numerical function g defined on K, and every ε > 0, there exist a polynomial h = P((f_n)) in the f_n, and a compact subset K_1 of K, such that \|h\| ≤ 2\|g\|, |\mu|(K − K_1) ≤ ε, and, for every x ∈ K_1, |h(x) − g(x)| ≤ ε. (Consider the mapping x ↦ (f_n(x)) of K into $\mathbf{R}^N$ and the closure of its image, and apply the Stone–Weierstrass theorem suitably.) From this, deduce that for 1 ≤ p < +∞, the set of polynomials in the f_n is dense in $\mathscr{L}^p$(K). Are these properties again true when the sequence (f_n) is replaced by an uncountable family of bounded measurable functions separating the points of K?