How can you measure neurotransmitter release from a presynaptic neuron by recording from a postsynaptic neuron

it would depend on the transmitter in question plus the tissue and the presence of enzyme environment degrading the neurotransmitter.....complex question.....neurophysiology prof would be better to elaborate on this

Easy, by measuring from the postsynaptic neuron, you can see how many neurotransmitters were actually received.  Knowing the rate of neurotransmitter response, release can be determined.  Besides, its more practical to do it this way.  The derivative of the change in rise time is the slope.  In other words, the slope describes the rate at which the change in rise time occurs.  Finally, there are several factors which could contribute to both the change in rise time and the rate of change.  Namely, change in pH and introdution of various bacteria.

To expand on the second person's answer you would be measuring the change in charge around the membrane of the postsynaptic neuron.  Neurotransmitters would either make it more positive leading to an action potential or more negative which would make an action potential less likely.  Since nerotransmitter affect channels or pumps that change ion locationsone of these two things is going to happen.
   The other reason to measure the change in action potential is that all you need is a volt meter. Otherwise you would need a device that could measure microscopic amounts of the 50+ neurotransmitters that occur in the body.  Some of which we haven't even discovered yet.

Easy.