Why am I getting strange results when using a power meter to measure CFLs?
Why your various meters don't do a good job measuring power being used by various devices is due to an AC Circuit phenomenon known as Power Factor.

Power Factor The ratio of real power to apparent power.

Real power is the work done, apparent power is the product of the current and voltage in the AC circuit. 

Capacitive and inductive loads cause a phase change between the current and voltage. 

Inductive loads cause current to lag voltage and capacitive loads cause current to lead voltage. Think of two sine waves, one representing current and the other representing voltage that get offset by up to 90 degrees depending on the type of load. This phase shift messes up the reading of the real power consumed by the circuit. And if the power meter does not measure power factor and correct for it, you basically have a good door stop.

If you have a lot of motors in a plant, you often have to add capacitors on the power line coming into the building so you don't mess the power grid up. It is done to reduce the phase difference between current and voltage.

So, if you're going to measure AC Power, the device must measure True Power (Watts), Apparent Power (VA), Power Factor (%PF) in addition to Voltage (RMS), Current (Amps) and Frequency (Hz). Once you have the True Power and Time, you can read kWH consumed.