We derive the optimal energy error estimate for multiscale finite element method with oversampling technique applying to elliptic system with rapidly oscillating periodic coefficients under the assumption that the coefficients are bounded and measurable, which may admit rough microstructures. As a by-product of the energy estimate, we derive the rate of convergence in L$^{d/(d-1)}-$norm.. The extension to locally periodic media will also be discussed. This is a joint work with Siqi Song.