But what if f is the following function:
f(g(x)) = \int^{b}_{a} g(x) dx
If that's the case,
f(lim_{n\rightarrow\infty} g_{n}(x)) = \int^{b}_{a} lim_{n\rightarrow\infty}g_{n}(x) dx
and
lim_{n\rightarrow\infty} f(g_{n}(x)) = lim_{n\rightarrow\infty} \int^{b}_{a} g_{n}(x) dx...
When is the following true?
f(lim_{n\rightarrow\infty}\ g_{n}(x)) =
lim_{n\rightarrow \infty}\ f(g_{n}(x))
Does anyone know of a textbook that discusses this?
This question is from Parzen (Modern Probability Theory), chapter 2, exercise 6.1
Homework Statement
Suppose that we have M urns, numbered 1 to M and M balls, numbered 1 to M. Let the balls be inserted randomly in the urns, with one ball in each urn. If a ball is put into the urn bearing...
I see my mistake. A constant is an upper function, but:
if u \in U(I) and f = u. Then f \in L(I). But \int -f = - \int u. The negative must be on the outside of the integral sign for u.
I am comparing theorem 10.6(c) and 10.14(a) in Apostol's Mathematical Analysis.
My question is this:
Are constant functions considered upper functions? They certainly seem to fit the definition 10.4 for upper functions:
A real-valued function f defined on an interval I is called an upper...
Homework Statement
(10.4 in Mathematical Analysis by Apostol)
This exercise gives an example of an upper function f on the interval I = [0,1] such that -f (not a member of) U(I). Let {r1, r2, ...} denote the set of rational numbers in [0,1] and let I_n = [r_n - 4^-n, r_n + 4^-n] (intersect)...