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Math Help - The Limit of Integrals Is the Integral of the Limit

  1. #1
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    The Limit of Integrals Is the Integral of the Limit

    I am asked to prove that, if $\displaystyle \lim_{n \rightarrow \infty} ||f_{n} - f|| = 0$ and $\displaystyle \lim_{n \rightarrow \infty} ||g_{n} - g|| = 0$, prove that  $\displaystyle \lim_{n \rightarrow \infty} \int_{I} f_{n} \cdot g_{n} = \int_{I}f \cdot g$.

    I doubt that I want to do this by brute force by providing some increasing sequence of upper functions that approach f \cdot g, though maybe I do since I know that \displaystyle \lim_{n \rightarrow \infty} ||f_{n}|| = ||f|| and likewise for g.

    In any case, the alternative seems to be to use to the Dominated Convergence Theorem, in which case I want to show that \{ f_{n} \cdot g_{n} \} converges almost everywhere (presumably to f \cdot g), and that there is a non-negative function which is integrable and dominates |f_{n} \cdot g_{n}|. But the first part seems like it might be false, since I don't know that the interval over which these functions are integrated is bounded.

    Maybe I want to try an epsilon-delta argument?
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  2. #2
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    What about uniform convergence? It seems that you are not given enough information though.
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  3. #3
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    Nope, I'm not given that the functions are uniformly convergent, just that they are in L(I) and measurable.
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  4. #4
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    Quote Originally Posted by ragnar View Post
    I am asked to prove that, if $\displaystyle \lim_{n \rightarrow \infty} ||f_{n} - f|| = 0$ and $\displaystyle \lim_{n \rightarrow \infty} ||g_{n} - g|| = 0$, prove that  $\displaystyle \lim_{n \rightarrow \infty} \int_{I} f_{n} \cdot g_{n} = \int_{I}f \cdot g$.

    I doubt that I want to do this by brute force by providing some increasing sequence of upper functions that approach f \cdot g, though maybe I do since I know that \displaystyle \lim_{n \rightarrow \infty} ||f_{n}|| = ||f|| and likewise for g.

    In any case, the alternative seems to be to use to the Dominated Convergence Theorem, in which case I want to show that \{ f_{n} \cdot g_{n} \} converges almost everywhere (presumably to f \cdot g), and that there is a non-negative function which is integrable and dominates |f_{n} \cdot g_{n}|. But the first part seems like it might be false, since I don't know that the interval over which these functions are integrated is bounded.

    Maybe I want to try an epsilon-delta argument?
    Actually you only need that one of the sequences converges in norm, the other can converge weakly and the result still holds (assuming you're working in L^2, otherwise the products f_ng_n and fg may not be integrable):

    \displaystyle| \int_I f_ng_n - \int_I fg | \leq |\int_I f_ng_n - \int_I f_ng | +| \int_I f_ng - \int fg| \leq \| f_n\| \| g_n-g\| + | \int_I f_ng - \int_I fg | \rightarrow 0.

    Notice then that we only require f_n to coverge weakly to f and g_n to converge in norm. The proof also shows that th result holds if f_n,f \in L^p and g_n,g\in L^q with \frac{1}{p} + \frac{1}{q} =1 and 1<p,q<\infty.
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