Dr. Conrad Murray's girlfriend is begging the courts to let him out of jail pending his appeal ... claiming he's a good dad to their toddler.
Nicole Alvarez filed a declaration of support for Murray today ... in which she says if Murray is released he will live with her and their son in Santa Monica.
It's not the most powerful or emotional letter. Alvarez says, "Dr. Murray is the father of my child" -- and adds ... he "has been our primary source of support."
It's unclear how much support Murray can provide now -- considering his medical license has been revoked.
Murray's hearing for bail pending appeal of his manslaughter conviction is scheduled for tomorrow morning in downtown L.A.