Do I have to invite my sister to the wedding?

I'm torn on whether or not I should invite one of my sisters. I'm currently more than dead to her, according to her. I haven't even told my Mom I'm engaged, because I kinda want to decide what to do about my sister first. The very short version of the backstory: She has a VERY long track record of lying, stealing from family (mostly my mom, but also returned things my mom bought my kids before Mom could send them and pocketed the &), never keeps a job long so sponges off everyone. One huge blowout between us ended with me not speaking to her 5+ years. She had faked having cancer starting about a year after us losing our Grandma to cancer. I'm now "dead" to her because about a year ago we got into a fight when my mom had a spinal collapse, was partially paralyzed, needed major back surgery and my sister had every excuse possible for not being with Mom (I'm hallway across the country and was at my 2nd day of a new job, while sister was unemployed and lived with my Mom). Continued.....