Denote by span {f 1,f 2, …} the collection of all finite linear combinations of the functionsf 1,f 2, … over ℝ. The principal result of the paper is the following. Theorem (Full Muntz 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 (λ j ) ∞ is a sequence of distinct real numbers greater than −(1/p). Then span {x λ1,x λ2,…} is dense in Lp(A) if and only if $$\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $$ . Moreover, if $$\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $$ , then every function from the Lp(A) closure of {x λ1,x λ2,…} can be represented as an analytic function on {z ∈ ℂ \ (−∞,0] : |z| < rA} restricted to A ∩ (0, rA) where $$r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Muntz, Szasz, Clarkson, Erdos, P. Borwein, Erdelyi, and Operstein. Related issues about the denseness of {x λ1,x λ2,…} are also considered.