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Suppose [tex]\alpha(x) = [x][/tex] is the floor function, then what is the value of [tex]\int_{0}^{n}f d \alpha[/tex]

for [tex]f(x) \in R_{\alpha}[0,n][/tex]

where n is an integer?

This was a question on my exam. I want to know if I got it right. Some say the answer is zero, but I think it is:

[tex]\sum_{i=1}^{n-1} \max \{ f(x) : i < x \leq i+1\}[/tex]

Because [tex]M_i = \sup \{ f(x) | x_{i-1} \leq x \leq x_i \}[/tex] gets multiplied with the difference of the endpoints of every possible partition and, with a fine enough partition, we will get the value 0 most of the time and we will get the vale 1 n-1 times.

BACKGROUND

Let me add some more info:

To take the Stietjes integral you write [tex]P = \{a= x_0 < x_1 < \cdots < x_n = b \}[/tex] a partition of the interval [a ,b]. Then [tex] \Delta \alpha_i = \alpha( x_i) - \alpha (x_{i-1})[/tex] for i= 1, ..., n. Next for each i = 1, ..., n we define:

[tex]m_i = \inf \{ f(x) : x_{i-1} \leq x \leq x_i \}[/tex]

[tex]M_i = \sup \{ f(x) : x_{i-1} \leq x \leq x_i \}[/tex]

Now we can define the lower and upper Stietjes sums:

[tex]L(f, P) = \sum_{i=1}^{n}m_i\Delta \alpha_i [/tex]

[tex]U(f, P) = \sum_{i=1}^{n}M_i\Delta \alpha_i [/tex]

Now we can define the lower and upper Stietjes integrals, which are equal for any Stietjes integrable function over a given [tex]\alpha[/tex].

[tex]\bar{\int_{a}^{b}}f d \alpha = \inf_P U(f,P) [/tex]

[tex]\int_{a}^{b}f d \alpha = \sup_P L(f,P) [/tex]

So that's what we are talking about with this problem.

for [tex]f(x) \in R_{\alpha}[0,n][/tex]

where n is an integer?

This was a question on my exam. I want to know if I got it right. Some say the answer is zero, but I think it is:

[tex]\sum_{i=1}^{n-1} \max \{ f(x) : i < x \leq i+1\}[/tex]

Because [tex]M_i = \sup \{ f(x) | x_{i-1} \leq x \leq x_i \}[/tex] gets multiplied with the difference of the endpoints of every possible partition and, with a fine enough partition, we will get the value 0 most of the time and we will get the vale 1 n-1 times.

BACKGROUND

Let me add some more info:

To take the Stietjes integral you write [tex]P = \{a= x_0 < x_1 < \cdots < x_n = b \}[/tex] a partition of the interval [a ,b]. Then [tex] \Delta \alpha_i = \alpha( x_i) - \alpha (x_{i-1})[/tex] for i= 1, ..., n. Next for each i = 1, ..., n we define:

[tex]m_i = \inf \{ f(x) : x_{i-1} \leq x \leq x_i \}[/tex]

[tex]M_i = \sup \{ f(x) : x_{i-1} \leq x \leq x_i \}[/tex]

Now we can define the lower and upper Stietjes sums:

[tex]L(f, P) = \sum_{i=1}^{n}m_i\Delta \alpha_i [/tex]

[tex]U(f, P) = \sum_{i=1}^{n}M_i\Delta \alpha_i [/tex]

Now we can define the lower and upper Stietjes integrals, which are equal for any Stietjes integrable function over a given [tex]\alpha[/tex].

[tex]\bar{\int_{a}^{b}}f d \alpha = \inf_P U(f,P) [/tex]

[tex]\int_{a}^{b}f d \alpha = \sup_P L(f,P) [/tex]

So that's what we are talking about with this problem.

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