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# Properties of another alternate definition of the integral of a step function

Suppose we defined the integral of a step function as:

Which of the following properties of the integral would still hold:

1. .
2. .
3. .
4. .
5. If for all , then .

1. True.
Proof. Let be a partition of and be partition of such that is constant on the open subintervals of and . Then, let , where , so is a partition of and is constant on the open subintervals of . Then,

2. True.
Proof. Let be partition of such that is constant on the open subintervals of . (We know such a partition exists since we can take the common refinement of the partitions of on which and individually are constant.) Assume if for . Then,

3. True.
Proof. Let be a partition of such that is constant on the open subintervals of . Assume if for . Then if so using our alternative definition of the integral we have,

4. False. Since . In particular, a counterexample is given by letting for all and let . Then,

5. False.
A counterexample is given by considering and on the interval . Then, on the interval, but