I grew up in Chicago, and although I've never been at a mall early morning, I've never seen any dogs that weren't service dogs inside.
My suggestion is to call the actual mall or store you have in question. I've read about Home Depot being dog friendly, but I called mine (and maybe got the wrong employee) and they told me that they only allow service dogs.
I guess I question why you're looking at such a public place as a "training ground." It really is unnecessary in my opinion to be toeing the law for such things when there are areas where no one would even bother to question you being there with your dog.
CD RE TC HIC 7/10