I am reading some notes, and at one point, given a filtered probability space on $[0,T]$, they define a new measure $Q$ on $\mathcal{F_t}$ by taking a variable $L \in \mathcal{F}_T$ and letting $dQ = LdP$.
My question is, what does this notation mean? $dQ = LdP$?
edit: Since I did not get an answer yet: taken literally, does it mean something like if I take some set $A$ and add a bit to it to get some set $B$, then $Q(B) - Q(A) = L ( P(B) - P(A))$?
This is the Radon Nikodym derivative which is used to change the probability measure. The measure change is a well known process used in finance for several purposes and the most "known" one is the measure change in the B&S framework which is used to get the martingale property of the actualized asset price. If you want more details on that I suggest you take a look on Girsanov theorem.
When you write $dP=LdQ$ this means that your measure $P$ the probability of an event A to happen $P(A)=E^P[1_A]=\int_{A}dP(x)=\int_{A}LdQ(x)=E^Q[1_AL]$ this can be useful when some processes do not have a property that we need under the P measure but they do under the Q measure.
The $L$ process must fulfill some conditions so it can represent a measure change for more details I suggest you take a look on Radon Nikodym derivative and on Novikov criterion for the Girsanov theorem application.