Theorem 6.17. (Chebyshev’s inequality.) If for , then for any ,

Proof. Let . Then

Theorem 6.18. (Schur’s test.) Let and be two -finite measure spaces. is measurable function on . Suppose that there exists such that

and

Let . If , then the integral

converges absolutely for , and the function thus defined is in and

Proof. Suppose that . Let be the conjugate exponent to , i.e., . We write

.

By applying the Holder inequality,

.

By Tonelli’s theorem,

Thus by Fubini’s theorem for the mixed exponents (in this case, )\bigr), for and so that is well defined , and

The proofs for are similar.

Theorem 6.19. (Minkowski’s inequality. ) Suppose that and be two measure spaces, and be an measurable function on .

(a). If and , then

(b). If , for , and the function is in , then for , the function is , and

Proof. Step 1. We first prove (a). If , (a) is just the Tonelli’s theorem.

If , let be the conjugate exponent to and suppose that . Then by Tonelli’s theorem and Holder’s inequality,

So by Theorem 6.14, (a) follows.

Step 2. When , (b) follows from (a) and the Fubini’s theorem.

When , it is obvious.

Definition. If is a measurable function on , we define its distribution function by

Proposition 6.22. (a). is decreasing and right continuous.

(b). If , then .

(c). If increases to , then increases to .

(d). If , then

Proof. (a). Let , then

And the latter sets are increasing. So (a) follows.

(b) is obvious.

(c). For any ,

(d). If , then or .

Proposition 6.24. If , then

.

Proof. For any ,

Then

Note that is an measurable function.

Proposition 6.23. If for all , and is a nonnegative Borel measurable function on , then

Proof. We note that is increasing and right continuous. The construction in Theorem 1.16 gives a nonnegative and unique Borel measure on such that

for all . is the Lebesgue-Stieltjes measure associated with .

When ,

.

.

This proves the equation for characteristic function of a interval. By the uniqueness in Theorem 1.14 and , the equation is true for a Borel set; so follows the simple functions. Then by Theorem 2.10, the general case follows by the monotone convergence theorem.

Theorem 6.24. Let be a measure on the Borel sets of the positive real line such that is finite for every . Note that and that being monotone, is Borel measurable. Let be a measure space and an nonnegative measurable function on . Then

.

In particular, by choosing for $p>0$, we have

Proof. We write

Since is a measurable function, Fubini’s theorem gives

.

However,

Definition. If is a measurable function on and , we define

The weak is defined to be

Remark. (a). but not equal. Example: on .

(b). is not a norm because the triangle inequality fails. Example: . and .

We compute that

.

So

and similarly for .

However

So So