This paper deals with the range of vector-valued measures. The main purpose is to clarify the well-known Lyapunov convexity theorem for vector-valued measures on a real interval \([a,b]\). A vector measure \(\mu=(\mu_1,\dots,\mu_n)\) on \([a,b]\) is an oriented measure if it is nonatomic and if for each \(k\)-tuple \((1\leq k\leq n)\) of disjoint measurable subsets \((A_1,\dots,A_k)\) such that \(A_1<\dots< A_k\) (that is \(x_1<\dots< x_k\) if \(x_i\in A_i\)), the determinant \(\text{det} (\mu_i(A_j)_{1\leq i,j\leq k})\) is positive. The main result is the following. If \(\mu\) is an oriented measure on \([a,b]\) and \(\rho\) a measurable function from \([a,b]\) with values in \([0,1]\), then there exist \(a=\alpha_0\leq \alpha_1\leq\dots\leq \alpha_n\leq \alpha_{n+1}=b\) and \(a=\beta_0\leq \beta_1\leq\dots\leq \beta_n\leq \beta_{n+1}=b\) such that \[ \mu\Bigl(\bigcup_{0\leq i\leq n,\;i\text{ even}}[\alpha_i,\alpha_{i+1}]\Bigr)= \int^b_a\rho d\mu=\mu\Bigl(\bigcup_{0\leq i\leq n,\;i\text{ odd}}[\beta_i,\beta_{i+1}]\Bigr). \] Applications to the range of oriented measures are next given. This paper gives different proofs and completes a previous paper by the authors [J. Funct. Anal. 126, No. 2, 476-505 (1994; Zbl 0828.49002)].