Does the measured mass of earth include that of the atmosphere? Just what the title states please... and perhaps naive too... 
Wikipedia pegs the mass of our Earth at 5.9722 × 10^24 kg.
Does this figure include the mass of the Atmosphere?
 A: The mass of the earth can be and has been measured using Newton's law of gravity. For the original experiment see this wiki article. The method by construction excludes most of the atmosphere, and the value is valid below the radius where it is measured.
Measurements from space include the atmosphere and I expect that this NASA experiment exploring mass differences will also be giving the mass of the earth including the atmosphere. In any case the mass of the atmosphere with respect to the rest of the globe is 

The average mass of the atmosphere is about 5 quadrillion (5×1015) tonnes or 1/1,200,000 the mass of Earth

One should worry only if one is measuring with accuracy greater than this, as in the following fascinating experiment:
 I found this NASA experiment, which confirms general relativity and uses mass differences.
A: The difference is (just) past the last digit of accuracy in your number, so it is impossible to tell.
You can estimate the fraction of the mass of the atmosphere by the ratio of the atmosphere's height to the radius of the earth (which gives the order of magnitude of the fraction of the Earth's volume in the atmosphere, about 10km/6000km, or 1/600), times the ratio of density of gas to ordinary solid (which is about 1/300). This gives 1 part in $10^{-5}$, and this is an overestimate, because the Earth's core is much denser than an ordinary solid because of the immense pressure. The mass figure you quote is correct to about 1 part in $10^{-5}$, so it is not clear if it includes the atmosphere or not, because this is a negligible error.
The mass of the Earth is found by measuring g, and the 1 part in $10^{-5}$ (or less) variations due to the atmosphere as you go up will be hard to distinguish from oblateness corrections, or just experimental error.
Atmospheric pressure
Multiplying standard atmospheric pressure by the surface area of the Earth gives the mass of the atmosphere as $5 \times 10^{18}Kg$, which is almost exactly 1 part in $10^{-6}$ of the mass of the Earth that you quote.
