These spaces are complete. (If one uses the Riemann integral instead, the space is not complete, which may be seen as a justification for Lebesgue's integration theory.) Concretely this means that for any sequence of Lebesgue-integrable functions f 1 , f 2 , … … , f n , … … with ‖ ‖ f n ‖ ‖ p < ∞ ∞ , satisfying the condition lim k , n → → ∞ ∞ ∫ ∫ Ω Ω | f k ( x ) − − f n ( x ) | p d μ μ ( x ) = 0 there exists a function f ( x ) belonging to the vector space L p ( Ω Ω ) such that lim k → → ∞ ∞ ∫ ∫ Ω Ω | f ( x ) − − f k ( x ) | p d μ μ ( x ) = 0.