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Math Help - To integrate static?

  1. #1
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    To integrate static?

    A curious question occurred to me, and I am not familiar enough with real analysis to answer it. Consider a real function f(x): [a,b]\to[c,d] defined as follows: for each element x\in[a,b], let f(x) be some random element from [c,d], a function obviously completely discontinuous. What would the value of \int_a^b f(x)dx be? It would make intuitive sense that the area under the curve is equal to the length of the interval, b-a, times the average value of the function in that interval. Therefore the answer to the question would be (b-a)\frac{c+d}2. But how to prove this rigorously? Is the function even integrable and by what definition?
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  2. #2
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    A correct statement could be: Let (f(x))_{x\in[a,b]} be a family of independent random variables uniformly distributed on [c,d]. What can be said about the (random variable) \int_a^b f(x)dx, if this is well-defined?

    As a matter of fact, I would bet that f is almost-surely not measurable, hence the integral wouldn't make sense.

    You can try another way, for instance as an approximation: for n\geq 0, define f_n:[0,1]\to [0,1] (or with a,b,c,d) to be a step function where the steps have width \frac{1}{n} and independent uniformly distributed heights in [0,1]. This won't converge to a function when n\to\infty. However, \int_0^1 f_n(t)dt\to_n \frac{1}{2} almost-surely by the law of large numbers, thus for large n this function f_n nearly satisfies what you said.

    By the way, there is a name for such a function f: it is called a white noise, you can look that up.
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  3. #3
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    Laurent: Thank you for the suggestion. I believe the following sketch is much closer to a formal proof.

    Define f_n:[0,1]\to[0,1] as follows: Divide the interval [0,1] into n subintervals \Delta_i of width \frac1n. For each of these subintervals, let f_n(\Delta_i) be a random element of [0,1]. Now define g by sorting these subintervals from shortest to tallest, g_n(\Delta_{i-1})\leq g_n(\Delta_i) for all i. It should be plain that \int_0^1 f_n(x)dx=\int_0^1 g_n(x)dx for all n.

    1. Define g as the limiting function of g_n, ( g=\lim_{n\to\infty} g_n), which is g(x)=x. Of course, \int_0^1 g(x)dx=\frac12.

    Define f as the limiting function of f_n, ( f=\lim_{n\to\infty} f_n)

    2. Since \int_0^1 f_n(x)dx=\int_0^1 g_n(x)dx for all n, \int_0^1 f(x)dx=\int_0^1 g(x)dx=\frac12.

    Are steps 1 and 2 valid?
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  4. #4
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    Quote Originally Posted by Media_Man View Post
    Laurent: Thank you for the suggestion. I believe the following sketch is much closer to a formal proof.

    Define f_n:[0,1]\to[0,1] as follows: Divide the interval [0,1] into n subintervals \Delta_i of width \frac1n. For each of these subintervals, let f_n(\Delta_i) be a random element of [0,1]. Now define g by sorting these subintervals from shortest to tallest, g_n(\Delta_{i-1})\leq g_n(\Delta_i) for all i. It should be plain that \int_0^1 f_n(x)dx=\int_0^1 g_n(x)dx for all n.

    1. Define g as the limiting function of g_n, ( g=\lim_{n\to\infty} g_n), which is g(x)=x. Of course, \int_0^1 g(x)dx=\frac12.

    Define f as the limiting function of f_n, ( f=\lim_{n\to\infty} f_n)

    2. Since \int_0^1 f_n(x)dx=\int_0^1 g_n(x)dx for all n, \int_0^1 f(x)dx=\int_0^1 g(x)dx=\frac12.

    Are steps 1 and 2 valid?
    I don't feel like this would be any more proof-like than what I did: you give no argument for the limits, as if they were obvious. Perhaps you should re-read my post. The fact that g_n converges to x\mapsto x almost surely (you need to specify this) is probably true but the proof is not immediate for sure. On the other hand, when I said that \int_0^1 f_n(t) dt\to\frac{1}{2} a.s. by the law of large numbers, that was immediate since \int_0^1 f_n(t)dt=\frac{1}{n}\sum_{i=1}^n f(\frac{i}{n}). Then the fact that f_n converges to a function f should also be justified and as a matter of fact, like I wrote, you just can't: the sequence (f_n)_n doesn't converge... I gave no justification since this should be clear: indeed, for any x, the sequence (f_n(x))_{n\geq 0} is a sequence of independent random variables uniformly distributed on [0,1]...
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