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In this paper we introduce and study a distance, Gromov-Hausdorff distance,which measures how two filtered A $A_{\infty}$ categories are far away eachother. In symplectic geometry the author associated a filtered$A_{\infty}$category, Fukaya category, to a finite set of Lagrangian submanifolds. TheGromov-Hausdorff distance then gives a new invariant of a finite set ofLagrangian submanifolds. One can estimate it by the Hofer distance ofHamiltonian diffeomorphisms needed to send one Lagrangain submanifold to theother. A motivation to introduce Gromov-Hausdorff distance is to obtain acertain completion of Fukaya category. If we have a sequence of sets ofLagrangian submanifolds, which is a Cauchy sequence in the sense of Hofermetric, then the associated filtered A infinity categories also form a Cauchysequence in Gromov-Hausdorff distance. In this paper we develop a theory toobtain an inductive limit of such a sequence of filtered $A_{\infty}$categories. In other words, we give an affirmative answer to [Fu5] Conjecture15.34.

Gromov-Hausdorff distance between filtered $A_{\infty}$ categories 1: Lagrangian Floer theory