Inheriting a Tenant mid Month

[Michael Salamone]

I am about to close on my first property this month, a duplex with one side already rented. I decided that I will allow the tenant to stay since they appear to be current on rent and their side is immaculate (I will probably start with month to month or 6 month lease terms to make sure they will still be current with me) 

My question is: Should I collect rent for the month I purchase the property? The closing is on the 9th, and I feel strange collecting rent mid month like that. But, I also feel like I shouldn't be like "yeah don't worry about it" and need to start firm and set the tone. Thoughts?