Let $M$ be a compact connected $C^\infty$ Riemannian manifold, let $m$ denote its Riemannian volume measure, and let $f: M \to M$ be a topologically transitive $C^{1+\alpha}$ Anosov diffeomorphism for some $\alpha \in (0,1]$. Let \begin{align*} TM = E_s \oplus E_u \end{align*} denote the continuous $Df$-invariant stable and unstable splitting, let $W_{u,\mathrm{loc}}(x)$ denote a local unstable manifold through $x \in M$, and let $m_{u,x}$ denote the Riemannian volume measure induced on $W_{u,\mathrm{loc}}(x)$. Then there exists a unique $f$-invariant Borel probability measure $\mu_{\mathrm{SRB}}$ on $M$ whose conditional measures, with respect to every measurable partition subordinate to local unstable manifolds equivalently in every local product box away from partition-boundary ambiguity sets, are absolutely continuous with respect to the corresponding induced measures $m_{u,x}$. This measure is ergodic. Moreover, its basin \begin{align*} B(\mu_{\mathrm{SRB}}) := \left\{x \in M : \frac{1}{N}\sum_{k=0}^{N-1} \varphi(f^k(x)) \to \int_M \varphi \, d\mu_{\mathrm{SRB}} \text{ for every } \varphi \in C^0(M;\mathbb{R})\right\} \end{align*} has full Riemannian volume: \begin{align*} m(B(\mu_{\mathrm{SRB}})) = m(M). \end{align*}