I'm trying to define the Hausdorff distance between two fuzzy sets in terms of non-fuzzy sets. Is this a viable definition? How can I show that this reduces to the Hausdorff Distance for non-fuzzy sets?
Let $X$ be a set and $I:= [0,1]$. A fuzzy set $A$ in $X$ is a set uniquely determined by its membership function $\mu_{A}: x \in X \to [0,1]$, where $\mu_{A}(x)$ is the point $x$'s ``grade of membership''
Let $A$ and $B$ be two nonempty fuzzy sets in some metric space $X$. We say the maximum membership of a fuzzy set $A$ is: $$\alpha^*=\sup_{x \in X}\{\mu_A(x)\}$$ We then define the non-fuzzy set $\mathcal A_{\sup}$ as follows: Fix $0<\epsilon <1$ $$\mathcal A_{\sup}=\{x\,|\,\mu_A(x)\geq \alpha^* - \epsilon\}$$ Let $\mathcal A_{\ell}$ be a nonempty non-fuzzy subset of $X$ such that $$\mathcal A_{\sup}\subset \mathcal A_{\ell}$$ and such that for any two fuzzy sets $A$ and $B$ $$\mathcal A_{\ell} = \mathcal B_{\ell} \iff \mathcal A_{\sup} = \mathcal B_{\sup}$$ We define the family of non-fuzzy sets $\mathcal A_{t}$, where $t\in[0,1]$, by $$\mathcal A_{t}=\left\{ \begin{array}{ll} \{x\,|\,\mu_A(x)\in[t,\alpha^*]\} & \text{if } t\leq \alpha^* \\ \mathcal A_\ell & \text{if }t > \alpha^* \end{array}\right. $$ Not that $\mathcal A_t=\mathcal A_{\sup}$ if $t=\alpha^*$ and that the second case cannot arise if $\alpha^*=1$. First, we will suppose that the membership functions only take on a discrete set of membership values $t_j$ where $j\in J$ and $J$ is an index set. Let $H(\mathcal A, \mathcal B)$ be the regular Hausdorff distance between $\mathcal A$ and $\mathcal B$.That is, $$ H(\mathcal A, \mathcal B) = \max\{ \sup_{a \in \mathcal A} \inf_{b \in \mathcal B} d(a,b), \sup_{b \in \mathcal B} \inf_{a \in \mathcal A} d(a,b)\}$$
We now define the fuzzy set distance to be $$\mathcal H(A,B)=\frac{\sum \limits_{j\in J} t_j H(\mathcal A_{t_j}, \mathcal B_{t_j})}{\sum \limits_{j\in J}t_j}$$
Now we will consider the case when $A$ and $B$ are continuous valued. Analogous to the discrete case above, we define the fuzzy set distance to be $$\mathcal H(A,B) =\frac{\displaystyle\int_0^1{tH(\mathcal A_t,\mathcal B_t)\mathrm{d}t}}{\displaystyle\int_0^1t\mathrm{d}t}=2\int_0^1tH(\mathcal A_t,\mathcal B_t)\mathrm{d}t$$
It is clear that our definition depends on how we choose $\mathcal A_{\ell}$. If a fuzzy set $A$ has a maximum membership value $\alpha^*=1$ then $\mathcal A_{\ell}$ does not need to be defined since $\mathcal A_{t}$ does not rely on it's definition. However, if $\alpha^*<1$, we can defined $\mathcal A_{\ell}$ by the union of $\mathcal A_{\sup}$ and $\{x_A\}$ ($\mathcal A_{\ell}=\mathcal A_{\sup}\cup \{x_A\}$). This single point may very well have a negligible effect on the distance by simply giving it negligable area. As long as we choose $\mathcal A_{\ell}$ consistently our results will be consistent.