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futurebird
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Suppose \alpha(x) = [x] is the floor function, then what is the value of \int_{0}^{n}f d \alpha
for f(x) \in R_{\alpha}[0,n]
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:
\sum_{i=1}^{n-1} \max \{ f(x) : i < x \leq i+1\}
Because M_i = \sup \{ f(x) | x_{i-1} \leq x \leq x_i \} 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.BACKGROUNDLet me add some more info:
To take the Stietjes integral you write P = \{a= x_0 < x_1 < \cdots < x_n = b \} a partition of the interval [a ,b]. Then \Delta \alpha_i = \alpha( x_i) - \alpha (x_{i-1}) for i= 1, ..., n. Next for each i = 1, ..., n we define:
m_i = \inf \{ f(x) : x_{i-1} \leq x \leq x_i \}
M_i = \sup \{ f(x) : x_{i-1} \leq x \leq x_i \}
Now we can define the lower and upper Stietjes sums:
L(f, P) = \sum_{i=1}^{n}m_i\Delta \alpha_i
U(f, P) = \sum_{i=1}^{n}M_i\Delta \alpha_i
Now we can define the lower and upper Stietjes integrals, which are equal for any Stietjes integrable function over a given \alpha.
\bar{\int_{a}^{b}}f d \alpha = \inf_P U(f,P)
\int_{a}^{b}f d \alpha = \sup_P L(f,P)
So that's what we are talking about with this problem.
for f(x) \in R_{\alpha}[0,n]
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:
\sum_{i=1}^{n-1} \max \{ f(x) : i < x \leq i+1\}
Because M_i = \sup \{ f(x) | x_{i-1} \leq x \leq x_i \} 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.BACKGROUNDLet me add some more info:
To take the Stietjes integral you write P = \{a= x_0 < x_1 < \cdots < x_n = b \} a partition of the interval [a ,b]. Then \Delta \alpha_i = \alpha( x_i) - \alpha (x_{i-1}) for i= 1, ..., n. Next for each i = 1, ..., n we define:
m_i = \inf \{ f(x) : x_{i-1} \leq x \leq x_i \}
M_i = \sup \{ f(x) : x_{i-1} \leq x \leq x_i \}
Now we can define the lower and upper Stietjes sums:
L(f, P) = \sum_{i=1}^{n}m_i\Delta \alpha_i
U(f, P) = \sum_{i=1}^{n}M_i\Delta \alpha_i
Now we can define the lower and upper Stietjes integrals, which are equal for any Stietjes integrable function over a given \alpha.
\bar{\int_{a}^{b}}f d \alpha = \inf_P U(f,P)
\int_{a}^{b}f d \alpha = \sup_P L(f,P)
So that's what we are talking about with this problem.
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