Abstract: We study the relationship between the Riemann integration and the continuity almost everywhere of vector-valued functions and the property of Lebesgue (that is, every Riemann integrable vector-valued function is continuous almost everywhere). We prove that l^p(1 < p < \infty ) and l^\infty do not possess the property of Lebesgue by constructing two counterexamples, Further, based on the existing literature, we show that l^1 have the property of Lebesgue.