Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 208 / 6 / b / Solution

For the off-diagonal moves, the Metropolis–Hastings acceptance probability givesThis expression is unchanged by interchanging $x$ and $y$. The rejected-move measure $\pi (dx)r(x){\delta }_x(dy)$ is supported on the diagonal and is also symmetric. Therefore the whole transition measure satisfieswhich proves detailed balance, including the atom that an off-diagonal density alone would omit. Consequently the Metropolis–Hastings algorithm kernel preserves the normalized target measure.