Increase of internal energy due to measurement

We argue that the internal energy, E = (H), of a macroscopic system in thermal equilibrium must increase if we measure an observable A which does not commute with the Hamiltonian, H. We derive an expression for calculating a lower bound for this increase in E. We then generalize the above result, and show that under certain conditions the expectation value (C) of an observable C should increase by the measurement of another observable A, if A and C do not commute.