Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset of the critical points of $M$. Consider the negative of the gradient vector field $V$, that is at each point we consider the $(x, f(x)) \in M$ we assign the vector $V(x,f(x))=(−\nabla f(x),−|\nabla f|^2)$. This generates a flow $F: M \times \mathbb{R} \rightarrow M$ on $M$ and a set of integral curves that travel along the manifold toward the critical points of $M$. Consider a critical point $c \in M$ and define the attractor set $A(c) = \{p \in M: \exists t \in \mathbb{R}, F(p, t) = c\}$ of $c$. Let $\mathcal{C}$ be the set of critical points with property $P$. Consider the set $\mathcal{A} = \bigcup_{c \in \mathcal{C}} A(c)$, the set of all points that travel along integral curves to critical points in $\mathcal{C}$. Let $\mu$ be the volume measure on $M$. How could I compute the ratio $\frac{\mu(\mathcal{A})}{\mu(M)}$ and possibly lower bound it by some constant?

Edit: Fixed vector field definition according to Ben's correction.

Edit: Assume that $f > 0$ everywhere and has at least one critical point. I'm interested in any way we can approach this problem, so make any further assumptions you'd like on $f$.