Prove that detailed balance implies invariance of the distribution P(x) under the Markov chain T.

Proving that detailed balance holds is often a key step when proving that a Markov chain Monte Carlo simulation will converge to the desired distribution. The Metropolis method satisfies detailed balance, for example. Detailed balance is not an essential condition, however, and we will see later that irreversible Markov chains can be useful in practice, because they may have different random walk properties.