 In the context of ergodic theory, ‘almost everywhere’ means, by definition, ‘everywhere in \(X\) except, perhaps, in a set of measure zero’. That is, whenever a claim is qualified as ‘almost everywhere’ it means that it could be false for some points in \(X\), but these taken together have measure zero. Now we are in a position to explain what the phrase means in the definition of ergodicity. As we have seen above, the time average (but not the space average!) depends on the initial condition. If we say that \(f^* = \bar{f}\) almost everywhere we mean that all those initial conditions for which it turns out to be the case that \(f^* \ne \bar{f}\) taken together form a set of measure zero—they are like a line in the plane.