We study properties of Lebesgue measure (the smooth volume form) on the sphere under the action of an analytic map R of degree d greater than 1. Classical results show Lebesgue ergodicity cannot occur if R is a polynomial, but many Lebesgue ergodic and weakly Bernoulli degree 2 maps can be constructed. We show where in parameter space ergodic, weakly Bernoulli, and exact rational maps must occur for one family of quadratic maps and generalize to other families and other degrees. We also mention recent results on maps of (arbitrary) degree d for which Lebesgue measure is equivalent to a one-sided Bernoulli shift of entropy log d and relate these results to a conjecture of Mane and Lyubich.