 As a remark, all of the measures formula_68 are defined on the product sigma algebra on their respective spaces, which (as mentioned before) is rather coarse.The measure formula_64 may sometimes be extended appropriately to a larger sigma algebra, if there is additional structure involved.Note that the original statement of the theorem is just a special case of this theorem with formula_70 for all formula_71, and formula_72 for formula_73.