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Denote by span{f1, f2, . . .} the collection of all finite linear combinations of the functions f1, f2, . . . over ℝ. The principal result of the paper is the following. Theorem (Full Müntz Theorem in Lp(A) for p ∈ (0, ∞) and for compact sets A ⊂ [0,1] with positive lower density at 0). Let A ⊂ [0,1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λj)∞j=1 is a sequence of distinct real numbers greater than -(1/p). Then span{xλ1, xλ2, . . .} is dense in Lp(A) if and only if ∑∞j=1 λj+(1/p)/(λj+(1/p))2+1 = ∞. Moreover, if ∑∞j=1 λj+(1/p)/(λj+(1/p))2+1 < ∞, then every function from the Lp(A) closure of span{xλ1, xλ2, . . .} can be represented as an analytic function on {z ∈ ℂ (-∞,0] : |z| < rA} restricted to A ∩ (0,rA), where rA := sup{y ∈ ℝ : m(A ∩ [y, ∞)) > 0} (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of span{xλ1, xλ2, . . .} are also considered.