Jennifer Hudson just arrived at a Chicago courthouse, where she's HOPING to see a judge sentence William Balfour to life in prison for murdering her family.

31-year-old Balfour was convicted May 11th of 1st degree murder in the deaths of Jennifer's mom Darnell Donerson, her brother Jason Hudson, and nephew Julian King. He was also convicted of home invasion and residential burglary.

Balfour committed the triple murder in 2008 -- after flying into a jealous rage because his ex-girlfriend (Jennifer's sister) had received a gift from her new boyfriend.

Balfour requested a new trial today, but the judge turned him down. He faces a life sentence.