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Keeping it properly mixed would require constant intervention. So much better to have a passive barrier such as a roof. What happens if you mix heavy and light gasses. The light gasses still rise to the top and escape. The best you can do with heavy gasses on the moon is to create a situation where people don't need to wear space suits, but still require gas masks to breath as the oxygen is always going to rise to the top and escape.

After putting some thought into it I have figured out that a lifting gas that is lighter than oxygen would be required for the microbes to “hold the O2 down”. The microbes have to be able to “beat” the oxygen back into the upper atmosphere for it to work. This could be done by microbes splitting H2O into H2 (probably binding the O to some thing else) and then using the H2 to beat the free O2 to the upper atmosphere where it then reacts with the H2 and free O2 to make water and sink back down in the atmosphere.

I was thinking about the system used to hold O2 down. Another thing that could be done besides using Hydrogen to get the microbes into the upper atmosphere is just having them live there naturally using photosynthesis. I figure this way the microbes could also scavenge H2O and CO2 and use them for fuel. The only problem with that is that photosynthesis splits the Oxygen out of water making O2. But if the microbe just reacted the O2 with a sugar or a lipid this would fix the problem. After a certain period of growth the microbe would divide asymmetrically (aka bud) the smaller bud would stay in the upper atmosphere as the parent collected more O2, CO2 and water gaining mass. Eventually it would sink in the atmosphere after budding several times until it reached the ground releasing it’s water and gases.

For low gravity worlds a solid roof may not be quite necessary ( solid roof = possibly, the entire astronomical body engulfed into a geodesic sphere ... ) ... it is enough to provide very steep temperature gradient at the tropopause - "high wall cold trap" ... This will make the upper atmosphere too thin, decreasing the amount of atoms which reach escape velocity...

Imagine zillions of aerostats flying in the low stratosphere. Each of the baloons is: solar powered refrigerator with radiator... The strato-fridges sucks in air at the normal for such earth_like hights temperatures of 200-220 K, and releases it cooled down with 50 to 100 K. The excess heat is by-force ( say Peltier-Zeebek effect used ) radiated as IR either upwards ( day-time ), or downwards ( night-time )... It could be IRaser pointing the beams exactly where necessary for precise weather control with very "high resolution" in time, area and volume.

The strato-fridges - colourchanging for manipulation of the albedo - solar influx.

Cause the temperature of one world is nothing more than function of the heat retention rate of it == the ratio/balance between the infaling / absorbed energy and its emisivity in IR

Very often we visualise under term roof kinda greenhouse nylon or metal or glass carpet, or envision passive structure , but when I say strao-fridge-baloons I mean more kinda spores of kinda plants, which collect hydrogen in a bag and possess different types of organs / organoids to do fridging, pointing of IR, reflecting, COMMUNICATING between eachother AND with the parent plants in order to provide activelly, USING the eventual excess of solar EM radiation, control over the atmosphere retention , insolation, weather and climate of the respected world...

Such systems are nothing more super-natural than the earth`s Gaia system, or more super-natural than the warm-bloodedness of the mammals.

Such life-like systems, either self-replicating or not are the key to put control onto the exponential processes.

It is quite easy to calculate how much kelvins should be the gases in the terraformed-moon`s exobase and tropopause in order it to be kept for as long as the sun shines.

I found a power point presentation which is on topic to this discussion by Nick Hoekzema (a NASA scientist): atmostalk.pdf

Note to the anally inclined: The following paragraphs are speculation.

First off Hoekzema calculated that for most worlds thermal escape, which is what I assumed caused the greatest amount atmospheric loss, actually might not be that important. Hoekzema calculated that worlds with escape velocities as small as 0.8km/s could possibly hold a CO2 atmosphere for billions of years if thermal escape were the only mechanism working.

Using Hoekzema’s equations I calculated that the Moon could possibly hold at atmosphere of SF6 for longer than the age of the solar system and also possibly SF6/O2 atmosphere for longer than the age of the solar system. Also apparently the weight and thickness of the SF6 atmosphere would help defeat the two major causes of O2 loss: Hydrodynamic outflow (simplistically an updraft), and sputtering (fast moving molecules hitting slower molecules causing a cascade of atoms leaving the planet/moon). Both effects rely on the average velocities of the molecules in a gas so the SF6 “shields” the O2 from the effects. To visualize what’s occurring imagine a pool table with pool balls. Except some balls are light and small and some are big an heavy. Now if you shoot one of the small balls across a table with mostly large balls rather than light balls fewer balls will go in the sockets because when the small ball hits a large ball the small ball will slow down a lot and the large ball will barely speed up at all. This is the effect that reduces sputtering and hydrodynamic outflow. Although it’s hard to doubt that some mechanism would still be required to hold the O2 down, but it may not have to be super aggressive. Even without SF6 an O2 atmosphere could possibly stay on the moon for 100 million years according to Hoekzema’s calculations. Even if his calculations are highly optamistic the moon could still possibly hold an SF6/O2 atmosphere for 100’s of millions of years, even longer with a mechanism to slow the escape of gases in place.

So says Gallup in this online video

That's a neat map, but how come its all jagged around the edges? What do you expect the climate to be like with a 1 bar atmosphere?

Ittiz , BEAUTIFULL pics!!!!!!
But
1. SF6 + moisture + atmospheric electricity = phosgen...
2. chemical gravitational differentiation - such world would lose water pretty quick
3. from where to get all these exa-tonnes of F?