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