I haven't dealt with these new mirrorless cameras very much, but it doesn't seem like you are saving much space over a high end point and shoot (Canon G12, et. al) or a budget DSLR (Canon T3i, et. al)
Here is a comparison between the G1x and the NEX-5N. It shows the G1X winning, though it isn't really accurate because they only compared the NEX BODY (without lens), so the G1X got a higher score even though the NEX is superior in almost every facet.
Canon G1X vs Sony NEX-5N - Our Analysis
Here are the G1X advantages:
Image stabilization: "NO" for the NEX, because IS is built into the lenses.
Built in Flash: NEX kit comes with a detachable flash - so that should have been a wash
Viewfinder: G1X wins here as its built in, NEX you can buy a view finder attachment I believe
Exposure time: 60s vs. 30s, G1X wins.
Everything else though, NEX by a landslide:
bigger sensor: APS-C vs. 4/3's
Faster: 10fps vs. 1.9fps
Image quality: 77 vs. 60
Higher resolution, better colors, lighter / smaller / thinner, more dynamic range, lower noise, etc. etc.
Use that website to compare all kinds of cameras and it easily breaks them down so you can see the facts.
Here is the T3i vs. the NEX-5N