Pigeon seems lost

[tinab]

About a week ago a white pigeon with red legs and eyes showed up behind my restaurant. She\He is quite dirty. It doesn't have a band. It can fly up to the roof and back down to the ground. It lets me get pretty near while it's eating but hasn't let me touch it. It just sits there all day and night. It seems lost and lonely. It doesn't make any cooing noises but listens intently when I speak to it. I've been feeding her oats and crumbs and just went to the feed store to get some real pigeon food. Should I just leave it out there or catch it and bring it inside. She seems skinnier than she was when she first got here.