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Math Help - Translation Invariance of Integral

  1. #1
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    Translation Invariance of Integral

    Theorem. Let  f be a real-valued function that is Riemann integrable on  [a,b] . For a fixed  k \in \mathbb{R} , let  g_k be the function  g_{k}(x) = f(x+k) where the domain of  g is appropriately chosen so that  x is in the domain of  g_k if and only if  x+k is in the domain of  f . Then  g_k is Riemann integrable on  [a-k, b-k] and  \smallint_{a-k}^{b-k} g_k = \smallint_{a}^{b} f .


    Proof. Let  \epsilon >0 . Choose  \delta >0 such that  \delta < \frac{\epsilon}{2} . So basically we consider  \mathcal{R}(g_k, P) = \sum_{i=1}^{n} g_{k}(x_{i}^{*})(x_{i}-x_{i-1}) ? And then show that  |\mathcal{R}(g_k,P)-\mathcal{R}(f,P)| < \epsilon ?
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  2. #2
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    Quote Originally Posted by manjohn12 View Post
    Theorem. Let  f be a real-valued function that is Riemann integrable on  [a,b] . For a fixed  k \in \mathbb{R} , let  g_k be the function  g_{k}(x) = f(x+k) where the domain of  g is appropriately chosen so that  x is in the domain of  g_k if and only if  x+k is in the domain of  f . Then  g_k is Riemann integrable on  [a-k, b-k] and  \smallint_{a-k}^{b-k} g_k = \smallint_{a}^{b} f .
    Let I = \smallint_a^b f we know that for any \epsilon > 0 there exists \delta > 0 so that |\mathcal{R}(f,P) - I| < \epsilon where P is any partition with ||P||<\delta and \mathcal{R}(f,P) is any associated Riemann sum. Let P_k be any partition of [a-k,b-k] with ||P_k|| < \delta. If \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1}) is some Riemann sum then it is equal to \sum_{j=1}^n f(x_j+k) [(x_j+k)-(x_{j-1}+k)], but this is a Riemann sum associated with f where the partition norm is less than \delta. Thus, \left| \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1}) - I \right| < \epsilon, so we see that g_k is integrable with the same value.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    Let I = \smallint_a^b f we know that for any \epsilon > 0 there exists \delta > 0 so that |\mathcal{R}(f,P) - I| < \epsilon where P is any partition with ||P||<\delta and \mathcal{R}(f,P) is any associated Riemann sum. Let P_k be any partition of [a-k,b-k] with ||P_k|| < \delta. If \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1}) is some Riemann sum then it is equal to \sum_{j=1}^n f(x_j+k) [(x_j+k)-(x_{j-1}+k)], but this is a Riemann sum associated with f where the partition norm is less than \delta. Thus, \left| \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1}) - I \right| < \epsilon, so we see that g_k is integrable with the same value.
    Would it still prove it if we said \left| \sum_{j=1}^n g_k(t_j)(x_j-x_{j-1}) - \mathcal{R}(f,P) \right| < \epsilon instead of  I ?
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