'soft' brakes

My son brought his car over complaining that the brakes were shuddering..

I pulled the front right wheel and found the caliper was frozen..

Replaced the caliper and bled the system to get the air out.

If i am in the car without it running, I can pump the brakes and they are nice and firm.

As soon as I start the car, the brake pedal goes all the way to the floor and, while I can pump it a few times and it sort of firms up, as soon as I stop it drops back down to the floor.

Needless to say if I drive the car and hit the brakes, the car stops... eventually..

I was initially thinking brake booster, but everything that I read on that subject states that a bad brake booster would make the brakes hard to push, not soft.

Any thoughts out there?

As usual, thanks in advance.

Ken

The prime suspect would be that there is still air in the system, so my suggestion would be to bleed the entire system again. You might consider trying gravity bleeding - very slow by highly reliable with no chance of pilot error.

Before I mount and balance tires on my Aveo, I'm planning to bleed the brake system to remove the air inside. Hope this will solve the spongy brake pedal.