Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as

\begin{equation} J(a,b) = 1 - \frac{||a\odot b||_1}{||a\odot b||_1+||a-b||_1} \end{equation} where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.