Xerxes

Can anybody help with this? I'm trying to do function spaces, and bumped up against the Lebesgue integral. My books give a pat on the head, and tell me not to worry over much about it, then proceed to tell me it's "zero almost everywhere". That, as a definition, is about as much use as a rubber hammer. I am very familiar with the Reimann integral, but can anybody gentle me through this?