No it won't and you never have a "DVD" which is 870MB.

DVD video is MPEG2 format specifically designed and encased to play back on standard DVD players throughout the world. If you downloaded (illegally?) a video off the Internet, it was probably a MPEG4 video of some sort. DivX and other encoders are commonly used to rip DVDs to a more 'internet friendly' format. But, there is significant image degradation throughout the process.

I typically recommend that if you are making legal and legitimate backups of your personal DVD collection for portable transportation on an iPod device, that you try AnyDVD from Slysoft and pair it with Handbrake. Handbrake includes a number of video presets which will match a specific file size and allow for near 1:1 quality to the DVD as well as significant compression for serious portability.

I've converted about 200 DVDs to play back on my iPod and my AppleTV. AppleTV versions are about 2GB while iPod versions are about 800MB.

Oh - and no, you wouldn't increase the quality. If you start with a lousy source, you get the same lousy final product. ALWAYS start with your own DVD, then go from there.