This paper is concerned with the algebraic Riccati equations (AREs) related to theH∞filtering problem. A necessary and sufficient condition for theH∞problem to be solvable is that theH∞ARE has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite. It is shown that such a stabilizing solution is a monotonically non-increasing convex function of the prescribedH∞norm bound γ. This property of theH∞ARE is very important for the analysis of the performance of theH∞filter. In this paper, the size of the set of allH∞filters is considered on the basis of the monotonicity of the above Riccati solution. It turns out that, under a certain condition, the degree of freedom of theH∞filter reduces a1 the optimalH∞norm bound. These results provide a guideline for selecting the value of γ Some numerical examples are included.