I am studying the concepts of ergicity in a dynamic system. Let $(M, \mathcal A)$ be a measurable space and $T: M \to M$ a measurable map. Given a probability measure $P$, we can say that $P$ or $T$ is ergodic if: $$P(A)= 0 \quad\hbox{ or } \quad P(A)=1, \quad \forall A, \,\, A = T^{-1}(A)$$ But I have seen some books that before giving the definition of ergidicity, They require that $P$ be a $T-$invariant (or $T$ to be $P$-invariant) in the following sense: $$P(T^{-1}(A))= P(A), \quad \forall A \in \mathcal A\label{I}\tag{I}$$ See, for example, this book.
However, it seems to me that according of this topic, the definition of (\ref{I}) is not necessary.
Question
What is the real need to demand (\ref{I}) ?
Why is this necessary?
You are completely right, the notion of ergodic measure does not require an invariant measure. This is particularly useful in statistical physics since sometimes it is convenient to consider noninvariant Gibbs measures. For example, with such general notion:
we can show that a measure is ergodic if and only if any invariant function is constant almost everywhere;
and so many times we can establish ergodicity (even of noninvariant measures) by looking at Fourier coefficients.
Nevertheless, it is true that the most interesting consequences come when the ergodic measure is also invariant. For example, for an ergodic invariant measure it follows from Birkhoff's ergodic theorem that the time average along almost all trajectories is equal to the space average.