Let $k \geq 1$ be an integer and let $f\colon \mathbb C\mathbb P^k \to \mathbb C\mathbb P^k$ be a holomorphic endomorphism of algebraic degree at least 2. Let $\nu$ be an invariant ergodic probability measure with positive Lyapunov exponents. In this talk, we will introduce a volume dimension of the measure $\nu$ which is equivalent to the Hausdorff dimension when $k =1$ but depends on the dynamics of the map when $k \geq 2$ to incorporate the non-conformality of holomorphic maps in higher dimensional projective spaces. We prove a generalized Mane-Manning formula relating the dimension, entropy and Lyapunov exponents of $\nu$. As applications we will characterize the first zero of a pressure function for expanding invariant measure in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. The talk is based on joint work with Fabrizio Bianchi.