What does it mean to say a function is differentiable with respect to lebesgue measure? What does it mean to say a function is differentiable with respect to the lebesgue measure almost everywhere. A definition would be helpfull.

Do I need to learn about the Radon Nikodym derivative to understand this.  

Thanks
After some thought and since I didn't really get any satisfactory answers. This is what I make of the statement.

The definition of differentiation in the statement is taken in the usual sense (the standard defintion from calculus, for some reason I thought the statement was referring to some new definition of differentiation!). So the statemtnt simply says, The set of points at which the function is not differentiable is a null set w.r.t. the lebesgue measure.

So the statement would have been clearer I think if the author had written 

$f$ is differentiable almost everywhere with respect to the lebesgue measure ... I think thats right!!

So no new definition of differentiation, phew! just some confusing terminology.
I wonder what part of your question is not covered by this page.