How is the voltage across a capacitor measured in a passive LPF circuit with no ground?
I assume with ground, you mean a reference. All voltages are measured between two points. However, to make things easier we might want to choose one fixed point we refer all other measurements to. This is what we then often call "ground"

You see, when we talk about ground in a circuit, it's not some special point that is determined by some laws of physics. It is what we choose it to be. We say "This is what I will say is zero, and I will refer my other voltages to this point". Of course, there are often points that make sense to call ground: For example, the middle between two supplies, to give you a \$ \pm 12 \ V\$  supply. But we can equally call the most negative point the ground, and then we have a supply that gives us 12 and 24 V. 

Hence, the "absence" of a ground does not force any voltages. 

In your example. I assume there is a switch somewhere that closes at \$ t = 0 \$. Indeed, then we must look at the voltage equations to see how the voltage on each node behaves. It would make sense to define the node that is shared as the ground - but we don't have to. Then indeed, in an ideal world there might be some voltage across the capacitor already that can't escape. This would set our starting conditions.

When your Professor says "it is \$-10\ V\$, does he mean that in a "assume the voltage across the capacitor is \$-10\ V\$ for \$t \leq 0 \$?