Member properties can be added to a Dimension by using the Business Modeler. Member properties allow you to add custom attributes to a dimension which can be used to create filters for PerforamancePoint Forms and Reports, provide a meaningful description for a dimension member ,logic in a business rule and more. However, in the current version of PerformancePoint (CTP2), once a member property is created it cannot be deleted. This lack of functionality is not a bug. It is documented in the PerformancePoint Planning Business Modeler Help file under the content heading "Predefined dimensions" for some of the dimensions. Specifically if you look at the Currency Dimension there is a table that tells you what types of user defined objects can be created, updated or deleted for that dimension. Unfortunately this is only mentioned in a few places but the ability to delete a user defined member property is not possible for all dimensions.
Of course once I found out that you could not delete a member property I submitted a feedback item to the Microsoft Connect site as a suggestion. Furthermore I asked them if it was possible to do anything in the "back end" of the application to delete the property. The response to changing the back end, like deleting the member property from the Analysis Service cube, is that it would potentially cause errors in the PerformancePoint Application. There would be "sync" issues between the Analysis Service Cube and the PerformancePoint meta data layer. However, the good news is that Microsoft has accepted this as a valid suggestion so the functionality could change in future releases of the product. It is also important to remember that this lack of functionality doesn't really impact a PerformancePoint Planning Application. If you spend a little more time on design you can eliminate having a lot of unused member properties in the Application.