Other woman in my house

Well...Not sure if he actually had the b-lls to do so, but my intuition tells me that while I was at work and husband was picking up things at the house for our son, he may have invited his "girlfriend" into my house.  Furthermore, while going to see our daughter at her new job, a place I also had worked for 10 years, he along with our son, took his girlfriend with him.  I live in a small town of 4000 people where everyone knows everyone.  I feel mortified that he not only had the nerve to show up to a place I know so many people with "her", but more so, I have to wonder how my kids felt to have thier father be so bold and down right ignorant about how they might feel. I feel as though my boundries are being tested; pushed and squeezed to the point where I need to establish boundries out right.  This is intimidating to me, cause I don't want to rock the boat since he's been very generous financially, but I certainly need to approach this. His reaction to him taking our kids to meet the other woman (read my 1st post) was to say he didn't know he needed permission from me to do so. This is what I'm dealing with.  How do I set boundries and why is it so difficult for him to see how this threatens me within the confines of my own home and small community? But more importantly, how can he not see how this may make our kids uncomfortable?