Regarding Upper and lower integral sets.

[nascentmind]

I am having some doubts in the definitions of the upper and lower integrals in apostol.

There is a statement saying "Let S denote the set of all numbers $_{a}$$\int$ $^{b}$ s(x) dx obtained as s runs through all step functions below f i.e. S = { $_{a}$$\int$ $^{b}$ s(x) dx | s < f} "

I did not get this. Shouldn't S be a singleton with a only a single element being the summation of the area of all the step functions below f ?

 

[Bacle2]

From what I get, the integral refers to the numerical value of the step function

satisfying the condition s<f. So for each such step function you get the associated

number, so you end with S as a collection of numbers.

 

[nascentmind]

From what I get, the integral refers to the numerical value of the step function

satisfying the condition s<f. So for each such step function you get the associated

number, so you end with S as a collection of numbers.

When I do a $\int$$_{a}$$^{b}$ s(x) dx I should have single number no? Even if he is considering different values for the step functions he should finally sum it up because the definition of integration says so right?

 

[Bacle2]

When I do a $\int$$_{a}$$^{b}$ s(x) dx I should have single number no? Even if he is considering different values for the step functions he should finally sum it up because the definition of integration says so right?

That sounds right; for each choice of step function you get a numerical value-- the

Riemann integral of the step function.