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Thread: Today's integration proof #1

  1. #1
    Math Engineering Student
    Krizalid's Avatar
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    Today's integration proof #1

    1. Let $\displaystyle f:[a,b]\to\mathbb R$ be a monotone function on the compact interval $\displaystyle [a,b].$ Prove that $\displaystyle f$ is integrable on $\displaystyle [a,b].$
    2. Let $\displaystyle f:[a,b]\to\mathbb R$ be a continuous function on the compact interval $\displaystyle [a,b].$ Prove that $\displaystyle f$ is integrable on $\displaystyle [a,b].$

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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Three things

    1. I am only currently finishing up Riemann-Stieltjes integration...I am still a novice. I would not even be posting if this were not for "fun"

    2.I assume that a knowledge of the defintion of the Riemann integral is knownw

    3. I assume you mean integrable in respect to x











    Quote Originally Posted by Krizalid View Post
    1. Let $\displaystyle f:[a,b]\to\mathbb R$ be a monotone function on the compact interval $\displaystyle [a,b].$ Prove that $\displaystyle f$ is integrable on $\displaystyle [a,b].$
    Let me prove a stronger result.

    Suppose that $\displaystyle \left|\sum_i \left\{M_i-m_i\right\}\right|\leqslant M$ then $\displaystyle f\in\mathfrak{R}$

    Proof: We must merely prove that given $\displaystyle \varepsilon>0$ we may find a partion $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle U\left(P,f\right)-L\left(P,f\right)<\varepsilon$.

    Now for each partition $\displaystyle P=\left\{a=x_0,x_1,\cdots,x_n=b\right\}$ of $\displaystyle [a,b]$ choose $\displaystyle \Delta x_i=\frac{b-a}{n}$ (note this can be done because x is continuous).

    So

    $\displaystyle \begin{aligned}U-L&=\sum_{i=1}^{n}\left[M_i-m_i\right]\Delta x_i\\
    &=\frac{b-a}{n}\sum_{i=1}^{n}\left[M_i-m_i\right]\\
    &\leqslant M\frac{b-a}{n}<\varepsilon\quad\blacksquare\end{aligned}$

    The last step is true for sufficiently large n.

    Now I will show that if $\displaystyle f$ is monotone then $\displaystyle \sum\left[M_i-m_i\right]$ is bounded:

    Without loss of generality assume that $\displaystyle f$ is montonically increasing. Then we can see that

    $\displaystyle M_i=\sup_{[x_{i-1},x_i]}f=f(x_i)$ and

    $\displaystyle m_i=\inf_{[x_{i-1},x_i]}f=f\left(x_{i-1}\right)$

    So

    $\displaystyle \begin{aligned}\sum_{i=1}^{n}\left[M_i-m_i\right]&=\sum_{i=1}^{n}\left[f(x_i)-f\left(x_{i-1}\right)\right]\\
    &=\left(f(x_1)-f(x_0)\right)+\left(f(x_2)-f(x_1)\right)+\cdots+\left(f(x_n)-f(x_n-1)\right)\\
    &=f(x_n)-f(x_0)\\
    &=f(b)-f(a)\end{aligned}$


    And because Riemann integration is only applicable to bounded functions we can see that if $\displaystyle f$ is montonically increasing that $\displaystyle \sum_{i}\left[M_i-m_i\right]$ is bounded. The proof for montonically decreasing functions is almost identical.

    Let $\displaystyle f:[a,b]\to\mathbb R$ be a continuous function on the compact interval $\displaystyle [a,b].$ Prove that $\displaystyle f$ is integrable on $\displaystyle [a,b].$
    Similarly we must just show that for any $\displaystyle \varepsilon>0$ there exists a corresponding partition $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle U\left(P,f\right)-L\left(P,f\right)<\varepsilon$.

    So for every $\displaystyle \varepsilon>0$ choose $\displaystyle \eta>0$ such that $\displaystyle \eta[b-a]<\varepsilon$. Now since continuity and uniform continuity are equivalent on compact spaces we may choose a $\displaystyle \delta>0$ such that $\displaystyle |x-y|<\delta\implies|f(x)-f(y)|<\eta {\color{blue}(*)}$. Now choose a partition $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle \max\left\{\Delta x_i\right\}<\delta$, so then

    $\displaystyle \begin{aligned}U-L&=\sum_{i=1}^{n}\left[M_i-m_i\right]\Delta x_i\\
    &\leqslant \eta\sum_{i=1}^{n}\Delta x_i{\color{red}(*)}\\
    &\overbrace{=}^{\text{telescopes}}\eta[b-a]<\varepsilon\quad\blacksquare\end{aligned}$

    $\displaystyle \color{red}(*)$ is justified by $\displaystyle \color{blue}(*)$
    Last edited by Mathstud28; Jan 6th 2009 at 10:28 PM. Reason: Typo and clarification
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    1. Let $\displaystyle f:[a,b]\to\mathbb R$ be a monotone function on the compact interval $\displaystyle [a,b].$ Prove that $\displaystyle f$ is integrable on $\displaystyle [a,b].$
    2. Let $\displaystyle f:[a,b]\to\mathbb R$ be a continuous function on the compact interval $\displaystyle [a,b].$ Prove that $\displaystyle f$ is integrable on $\displaystyle [a,b].$
    These are well-known and nice results from analysis.

    1)Without lose of generality say $\displaystyle f$ is non-decreasing. Notice that $\displaystyle f(a) \leq f(x) \leq f(b)$ for $\displaystyle x\in [a,b]$ therefore $\displaystyle |f| \leq M$ for some $\displaystyle M>0$ hence the function is bounded. Now for a partition $\displaystyle P = \{ x_0,x_1,...,x_n\}$ with $\displaystyle x_j < x_{j+1}$ where $\displaystyle 0\leq j\leq n-1$ and $\displaystyle x_0=a,x_n=b$ define $\displaystyle \Delta_j = x_j - x_{j-1}$ and $\displaystyle M_j = \sup \{ f(x) : x\in [x_{j-1},x_j]\}$ and $\displaystyle m_j = \inf \{ f(x) : x\in [x_{j-1},x_j] \}$. Let $\displaystyle \Delta = \max \{ \Delta_j : 1\leq j \leq n\}$. Let $\displaystyle U_P$ be the upper sum and $\displaystyle L_P$ be the lower sum. This means $\displaystyle U_P - L_P = \sum_{j=1}^n (M_j - m_j) \Delta_j $. Now $\displaystyle M_j = f(x_j)$ and $\displaystyle m_j = f(x_{j-1})$. Therefore, $\displaystyle U_P - L_P = \sum_{j=1}^n (f(x_j) - f(x_{j-1}) ) \Delta_j$. Therefore, $\displaystyle U_P - L_P \leq \sum_{j=1}^n (f(x_j) - f(x_{j-1})) \Delta = \Delta (f(x_n) - f(x_0)) = \Delta(f(b)-f(a))$. This means for $\displaystyle \epsilon > 0$ pick $\displaystyle \Delta < \frac{\epsilon}{f(b)-f(a)}$ (assuming that $\displaystyle f(b)\not = f(a)$, but if $\displaystyle f(b) = f(a)$ then the function is constant!) Therefore, $\displaystyle U_P - L_P < \epsilon$ which forces $\displaystyle f$ be to integrable by Cauchy's criterion.

    2)Here we use the fact that $\displaystyle f$ is uniformly continous. Thus, for any $\displaystyle \epsilon > 0$ there is $\displaystyle \delta > 0$ so that if $\displaystyle |x-y| < \delta \text{ and }x,y\in [a,b]\implies |f(x)-f(y)|<\epsilon$. Let $\displaystyle P$ be a partition $\displaystyle \Delta < \delta$. Now $\displaystyle M_j = \max \{ f(x) : x\in [x_{j-1},x_j] \}$ and $\displaystyle m_j = \min \{ f(x) : x\in [x_{j-1},x_j]\}$ by extreme-value theorem. Consequently, $\displaystyle M_j - m_j < \epsilon$ since $\displaystyle \Delta_j \leq \Delta < \delta$. Thus, $\displaystyle U_P - L_P = \sum_{j=1}^n (M_j - m_j) \Delta_j \leq \sum_{j=1}^n \epsilon \Delta_j = \epsilon (b-a)$. And since this is true for any $\displaystyle \epsilon > 0$ it means we can make $\displaystyle U_P - L_P$ arbitrary small and it is integrable by Cauchy's criterion.
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