I agree with ProShooter...
If the property owner has a condition attached to his permission to enter, such as a business open to the public but not open to those who carry guns because those who carry guns do not have permission to enter, evidenced by a no guns sign and a person carrying a gun enters anyway then that person has engaged in the activity of being on/in private property without the owner's permission.... an activity commonly referred to as "trespassing".
Being asked to leave (really just politely being required to leave) actually means the person has been caught already in the act of being on/in the property without permission. But... different States have different laws concerning
when any legal penalties begin for the act of trespass.
For example... one State might require signs of a certain size and wording before a sign serves as legal notice. If that is so then a person is guilty of trespass if the sign is ignored and maybe even if the sign wasn't seen but a reasonable person should have seen it. Yet another State might consider a hand written sign as valid notification. While a different State's laws might require verbal notice by being asked to leave.
Thing is... just because a no guns sign has no force of firearm law behind it doesn't mean it might not have the force of trespass law behind it. And it is up to us to not only know the firearm laws but also any other laws that aren't firearm law but still might affect carrying a firearm of the State we happen to be in whether by residence or by visiting.
However, let us not confuse when the act begins with when the legal penalties for the act begin. The act of trespass begins when a person is in/on private property (that includes businesses) without permission. That is the definition of trespass.
http://definitions.uslegal.com/t/trespass/
Trespass Law & Legal Definition
Trespass is entering another person's property without permission of the owner or legal authority.
-snip-
-----------
After all ... a person who steals money from the cash register is still guilty of engaging in theft whether he is notified or asked not to steal/gets caught or not. And so it is with a person who trespasses... whether they get caught (being asked to leave) or not they are still engaging in trespassing because they don't have the owner's permission. When the legal penalties kick in, including what serves as legal notice, differs with the different laws of different States.