Exercise 7.4.6

Although not part of Theorem 7.4.2, it is true that the product of integrable functions is integrable. Provide the details for each step in the following proof of this fact:

(a)
If f satisfies | f ( x ) | ≤ M on [ a , b ] , show | ( f ( x ) ) 2 − ( f ( y ) ) 2 | ≤ 2 M | f ( x ) − f ( y ) |

(b)
Prove that if f is integrable on [ a , b ] , then so is f 2 .
(c)
Now show that if f and g are integrable, then fg is integrable. (Consider ( f + g ) 2 .)

Answers

(a)
| f ( x ) 2 − f ( y ) 2 | = | f ( x ) + f ( y ) | | f ( x ) − f ( y ) | ≤ 2 M | f ( x ) − f ( y ) |
(b)
Consider some subinterval [ c , d ] ⊆ [ a , b ] , with:
m k = inf ⁡ { f ( x ) : x ∈ [ c , d ] }
M k = sup ⁡ { f ( x ) : x ∈ [ c , d ] }
m k ′ = inf ⁡ { f ( x ) 2 : x ∈ [ c , d ] }
M k ′ = sup ⁡ { f ( x ) 2 : x ∈ [ c , d ] }

We have that M k ′ − m k ′ ≤ 2 M ( M k − m k ) ; this implies that for any partition P ,

U ( f 2 , P ) − L ( f 2 , P ) ≤ 2 M ( U ( f , P ) − L ( f , P ) )

which since M is constant, implies f 2 is integrable.

(c)
If f and g are integrable, then so are f + g , ( f + g ) 2 = f 2 + 2 fg + g 2 , 2 fg , and fg .
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2022-01-27 00:00
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