Ok... how about I just try to make this "trespass" thing as simple as possible. May I strongly suggest following the link below and reading the information provided. Although I must caution folks to search
the actual black letter laws, firearm and trespass!, of their own State to be sure they understand all the nuances of those laws....
(the word "you" in the following is used in the generic sense)
Trespass | LII / Legal Information Institute
Trespass
Trespass is defined by the act of knowingly entering another person’s property without permission. Such action is held to infringe upon a property owner’s legal right to enjoy the benefits of ownership.
-snip-
Please note that the act of trespass is knowingly entering another person's property without permission. Please also note that whether a person "knows" or not they still don't have the owner's "permission".... but the "knowingly" is only a part of being legally guilty of "trespass".
It is the "entering" that constitutes "trespass" ... NOT the "being asked to leave". "Being asked to leave" means you got caught already engaging in the act of having entered without permission. Refusing to leave is merely continuing the act of trespass that you were already doing before you got caught ... but refusing to leave means the legal punishment for trespass will likely be suffered.
A sign of some kind, perhaps a list of things, that the property owner prohibits on/in his property is notice that those who engage in those prohibited things do NOT have the property owner's permission to be on/in his property. That sign tells you who does not have permission.
Now a person could actually not know/didn't see the sign (some businesses have signs that are off to the side and almost impossible to see) and the first they know about it is when they are asked to leave..... upon which time they now know they do not have the owner's permission to be there. Or they can lie (to themselves and/or to the owner) and say they didn't know (knowingly) when they get caught and are asked to leave (oh... make no mistake "asked to leave" really means "politely
required to leave").
The thing is... it doesn't matter what object the property owner says people are not allowed to bring in with them.... (some stadiums do not allow people to bring in coolers full of refreshments because they want to make money selling those refreshments... and some businesses do not allow people to bring in guns because they don't like guns/don't trust those folks who carry guns).... it still comes down to the simple fact that anyone who brings in a prohibited item does not have the owner's permission to be on/in the property and because they do not have permission they are therefor....
trespassing.
Again... first comes the trespass by a person sneaking in a gun where the property owner does not allow people with guns... then comes the getting caught part when the property owner asks (requires!) the person to leave... then comes the part where the person might suffer the legal punishment provided by the trespass laws.
But .... again....
first comes the trespass part by the person who is sneaking in the gun/cooler/whatever item the property owner has banned from his property.
By the way... I intentionally used the word "sneaking" because intentionally taking a concealed gun into a business where the person carrying knows the gun is not allowed IS........... "sneaking" the gun in. Although some people think "concealed means concealed" means as long as no one knows about it you can get away with it.... which would be "sneaking".
In the interest of full disclosure... I am not an attorney and the above is NOT legal advice... which is why I strongly suggest folks learn the applicable laws, both firearm and trespass and how they interact, (the actual black letter laws and not just what they hear from friends/cops or what they read in postings on the internet) in their own State of residence or States they intend to visit.