You need to check with your state. In Washington if your going to incorporate you can not do so under another businesses name. If you are a sole proprietor you have a tax id number but you can have your company listed DBA ie. Jane Smith DBA (doing business as) Pilates 1. Hope this helps again check your local laws.
One of the best things to do is go to http://www.uspto.gov and do a free Trademark search. If something comes up as a match look to see if it has anything to do with the fitness industry. If not you may be able to use it but I recommend talking with a Patent and Trademark attorney first. I would also do extensive Google searches on the name(s) you wish to use to find out more about who may be using the name already. Look for an (R) or (TM) next to any of these names which will be a red flag. Be absolutely sure to do your homework on this first before buying domain names, putting up a website, having printed material done and so on. Good luck!