The paper by R.J. DiPerna and P.L. Lions discusses new existence, uniqueness, and stability results for ordinary differential equations (ODEs) with coefficients in Sobolev spaces. These results are derived from corresponding findings on linear transport equations, analyzed using the method of renormalized solutions. The introduction highlights the Cauchy-Lipschitz theorem, which provides global solutions to autonomous ODEs with Lipschitz continuous vector fields. The authors extend this theory to vector fields in Sobolev spaces, providing additional information on the continuity and stability of solutions. They also derive estimates involving the Lebesgue measure and the image measure under the flow generated by the vector field, emphasizing the role of measure theory in this context.The paper by R.J. DiPerna and P.L. Lions discusses new existence, uniqueness, and stability results for ordinary differential equations (ODEs) with coefficients in Sobolev spaces. These results are derived from corresponding findings on linear transport equations, analyzed using the method of renormalized solutions. The introduction highlights the Cauchy-Lipschitz theorem, which provides global solutions to autonomous ODEs with Lipschitz continuous vector fields. The authors extend this theory to vector fields in Sobolev spaces, providing additional information on the continuity and stability of solutions. They also derive estimates involving the Lebesgue measure and the image measure under the flow generated by the vector field, emphasizing the role of measure theory in this context.