Simple LOD script problem (switching mesh at runtime)

I have a simple LOD script and it's not working. The problem I'm having is: the script is telling the mesh filter to change the mesh to a variable from the Lod script and the mesh filter is not obeying. Well anyway here's the script:

 public var mainCamera : Camera;
 public var lowMesh : Mesh;
 public var midMesh : Mesh;
 public var highMesh : Mesh;
 public var goneDistance : int = 500;
 public var lod0distance : int = 10;
 public var lod1distance : int = 20;
 public var lod2distance : int = 30;
 
 
 
 
 function Update () {
 
     var distance = Vector3.Distance(mainCamera.transform.position, this.transform.position);
 
     var mesh : Mesh = GetComponent(MeshFilter).sharedMesh;
     
 
     if (distance > lod1distance)
     {
         print ("Distance is greater than lod 1 distance.");
         if (midMesh)
         {
             mesh = midMesh;
             print ("Switched to Mid Mesh");
         }
 
     }
     if (distance > lod2distance)
     {
         if (lowMesh)
         {
             mesh = lowMesh;
             print ("Switched to Low Mesh");
         }
     }
     
     if (distance > goneDistance)
     {
         renderer.enabled = false;
     }
     else
     {
         renderer.enabled = true;
     }
 }

Here's what it looks like in the Inspector:

By the way, those variables with the values "head" are not all the same, they just have the same name.

And I am getting the messages, Switched to Mid Mesh and Switched to Low Mesh, etc. But the mesh is staying high poly. Any ideas? Thanks!