Apparatus for permanent placement across an ostium of a left atrial appendage in a patient, which includes a filtering membrane configured to extend across the ostium of the left atrial appendage. The filtering membrane has a permeable structure which allows blood to flow through but substantially inhibits thrombus from passing therethrough. The apparatus also includes a support structure comprising a plurality of fingers which are radially outwardly expandable with respect to a longitudinal axis to permanently engage the interior wall of the left atrial appendage, wherein the filtering membrane is attached to the support structure extending across the ostium of the left atrial appendage.