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Math Help - Lebesgue integrable

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    Lebesgue integrable

    I'm trying to prove f(x) = e^{-x^2} \in L^1(\mathbb{R}) ; that is, that f is Lebesgue integrable over \mathbb{R}.

    Let f(x) = e^{-x^2} \chi_{(-\infty , \infty)}(x) and f_n = f \chi_{[-n,n]}.

    Since f\geqslant 0, (f_n) is an increasing sequence of functions which converges everywhere to f.

    We want to show that f is integrable so it suffices to show \int f_n converges as n\to\infty. It then follows from the Monotone Convergence Theorem that f\in L^1 (\mathbb{R}) (and \int f = \lim_{n\to\infty} \int f_n).

    We use the given hint ("don't try to find \int f ; integrate a simpler upper bound instead").

    We see that f_n is bounded by ___. The integral of this upper bound converges, so by comparison test \int f_n converges.

    My hurdle is filling the gap for an upper bound. f_n is bounded by the constant function 1 but \int_{-n}^n 1 dx= 2n which doesn't converge as n \to\infty. Any suggestions?
    Last edited by ProofbyInduction; April 16th 2012 at 07:26 AM.
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