I think this might give some odd results when browsing the cube. However, it should be do-able.
I would probably keep a natural calendar week as part of the dimension, where weeks always start on Monday(?) and end on Sunday(?).
If I understand your statement you want to end up with partial weeks, not start the weeks on a different day of the week when the months transition. I would probably do something like a week of month column and a start date for week of month column. The week of month would be populated with 1, 2, 3, 4, 5. The start date for the week of the month column would be populated with something like 1-Sep-10, 6-Sep-10, 13-Sep-10, 20-Sep-10 and 27-Sep-10. I would probably also include, depending on your needs, a column that indicates the number of days in that month week. So the first week of September would have 5 as the number of days in the month week #1.
Hope this helps.