Today we will show more properties of integrals of simple functions. But the neat thing is that they will follow from the last two properties we showed yesterday. And so their proofs really have nothing to do with simple functions. We will be able to point back to this post once we establish the same basic linearity and order properties for the integrals of wider and wider classes of functions.
First up: if and are integrable simple functions with a.e. then
Indeed, the function is nonnegative a.e., and so we conclude that
Next, if and are integrable simple functions then
Here we use the triangle inequality and invoke the previous result.
Now, if is an integrable simple function then
The absolute value is greater than both and , and so we find
which implies the inequality we asserted.
As a heuristic, this last result is sort of like the triangle inequality to the extent that the integral is like a sum; adding inside the absolute value gives a smaller result than adding outside the absolute value. However, we have to be careful here; the integral we’re working with is not the limit of a sum like the Riemann integral was. In fact, we have no reason yet to believe that this integral and the Riemann integral have all that much to do with each other. But that shouldn’t stop us from using this analogy to remember the result.
Finally, if is an integrable simple function, is a measurable set, and and are real numbers so that for almost all , then
Indeed, the assumed inequality is equivalent to the assertion that a.e., and so — as long as — we conclude that
which is equivalent to the above. On the other hand, if , then must be zero on all but a portion of of finite measure or else it wouldn’t be integrable. Thus, in order for the assumed inequalities to hold, we must have and . The asserted inequalities are then all but tautological.