I have a clear definition for an ergodic endomorphism, but I can't seem to find a clear definition for an ergodic measure.
For example, the Wolfram definition (https://mathworld.wolfram.com/ErgodicMeasure.html) is more a definition of an ergodic endomorphism, not ergodic measure. Thanks
Ergodicity of measures is defined with respect with to an endomorphism, just like in the reference you provided. The idea is that the space on which your transformation acts cannot be divided into two non trivial parts (parts of positive measure) in which the transformation acts without leaving each part. You can see that both the measure and the endomorphism are essential for the definition. In general you say that the endomorphism is ergodic with respect to the measure or vice versa.