By definition is convex if and only if
whenever and are in the domain of .
It follows by induction on that if for then
Jensen's inequality says this:
If is a probability measure on ,
is a real-valued function on ,
is integrable, and
is convex on the range of then
Proof 1: By some limiting argument we can assume that is simple. (This limiting argument is a missing detail to this proof...)
That is, is the disjoint union of and is constant on each .
Say and is the value of on .
Then (1) and (2) say exactly the same thing. QED.
Lemma. If and then
The lemma shows:
- has a right-hand derivative at every point, and
- the graph of lies above the "tangent" line through any point on the graph with slope equal to the right derivative.
Let be the right derivative of at , and let
The bullets above say for all in the domain of . So
David C. Ullrich