Changing measure to speed up Monte Carlo

What does a change of probability measure do, and how can it be put to work inside a Monte Carlo pricer?

Reweighting paths by a Radon-Nikodym derivative. Changing measure means simulating under a different probability $\mathbb{Q}$ and multiplying each payoff by $d\mathbb{P}/d\mathbb{Q}$, so the expectation is unchanged (Girsanov describes how Brownian drifts shift). The classic Monte Carlo use is importance sampling: for a deep out-of-the-money option, drift the paths towards the strike so most finish in the money, then correct with the likelihood ratio, which can cut variance dramatically. Pricing under the risk-neutral or a numeraire measure is the same idea.