Another Dead Man Referred to Prosecutors for Murder

No matter how many times I read about these cases of dead people being prosecuted in Japan, I still feel there is something very strange. 

Police in Toyokawa City in Aichi Prefecture decided to send a recommendation to prosecutors to prosecute a dead man suspected of killing his girlfriend six months ago when he was alive.

On January 5th, police responded to an emergency call from Kurumi Baba, a 26 year old woman, crying for help. When police arrived, they found her collapsed and bleeding from stab wounds. She was transported to the hospital, but pronounced dead two hours later. Yu Nakayama, a man in his 30’s who was in a relationship with Kurumi Baba, was found with a neck wound. He was also transported to the hospital where he later died. Police believe Nakayama stabbed Baba, then attempted suicide by stabbing himself. A blood stained knife was found near him. 

I understand people wanting closure, but with both people dead, it seems pretty closed. Prosecution and trials are expensive. Is this the best use of public funds? What do you think?