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Thread: Integrability

  1. #1
    Senior Member I-Think's Avatar
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    Integrability

    Using these 2 facts for integrable functions f, g
    $\displaystyle 1. \int_a^b (f+g)=\int_a^b f+\int_a^b g
    $

    $\displaystyle 2. \int_a^b cf=c\int_a^b f $for a constant $\displaystyle c$

    Prove that any integrable function f can be changed at a finite number of points in the interval [a,b] without changing its integrability or its integral.

    Start of Proof
    I know that I only have to prove it for changing 1 point, and an induction argument will take care of the rest.
    But how do I go about doing this?

    I started with the definition of integrability
    $\displaystyle sup{ }L(f,P)=inf{ }U(f,Q)$, where $\displaystyle P $ and $\displaystyle Q $ are partitions of $\displaystyle [a,b]$

    and so dealing with only the sup and a partition $\displaystyle Pa<x_1<b)$ and $\displaystyle Qa<x_2<b)$
    I need to prove
    $\displaystyle sup{ }f[a,x_1](x_1-a)+sup{ }f[x_1,b](b-x_1)=sup{ }f[a,x_2](x_2-a)+sup{ }f[x_2,b](b-x_2)
    $
    How do I use the given facts to solve this problem. Or do I need a new approach
    Thanks in advance.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by I-Think View Post
    Using these 2 facts for integrable functions f, g
    $\displaystyle 1. \int_a^b (f+g)=\int_a^b f+\int_a^b g
    $

    $\displaystyle 2. \int_a^b cf=c\int_a^b f $for a constant $\displaystyle c$

    Prove that any integrable function f can be changed at a finite number of points in the interval [a,b] without changing its integrability or its integral.

    Start of Proof
    I know that I only have to prove it for changing 1 point, and an induction argument will take care of the rest.
    But how do I go about doing this?

    I started with the definition of integrability
    $\displaystyle sup{ }L(f,P)=inf{ }U(f,Q)$, where $\displaystyle P $ and $\displaystyle Q $ are partitions of $\displaystyle [a,b]$

    and so dealing with only the sup and a partition $\displaystyle Pa<x_1<b)$ and $\displaystyle Qa<x_2<b)$
    I need to prove
    $\displaystyle sup{ }f[a,x_1](x_1-a)+sup{ }f[x_1,b](b-x_1)=sup{ }f[a,x_2](x_2-a)+sup{ }f[x_2,b](b-x_2)
    $
    How do I use the given facts to solve this problem. Or do I need a new approach
    Thanks in advance.
    I would assume the intended method was as follows. I'll assume for ease that $\displaystyle \alpha$ is continuous. You can generalize


    Lemma: Let $\displaystyle c\in[a,b]$ and $\displaystyle \mathbf{1}_c$ the indicator function on $\displaystyle \{c\}$. Then, $\displaystyle \mathbf{1}\in\mathcal{R}(\alpha)$ on $\displaystyle [a,b]$ and $\displaystyle \displaystyle \int_a^b\mathbf{1}_c\text{ }d\alpha(x)=0$.
    Proof: Note that evidently $\displaystyle \displaystyle L(P,\mathbf{1}_c,\alpha)=0$ for every partition $\displaystyle P$ of $\displaystyle [a,b]$. Thus, it suffices to show that for every $\displaystyle \varepsilon>0$ there exists some partition $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle U\left(P,\mathbf{1}_c,\alpha\right)<\varepsilon$. To do this choose $\displaystyle \displaystyle x_1,x_2$ such that $\displaystyle \alpha(x_2)-\alpha(x_1)<\varepsilon$ and $\displaystyle c\in(x_1,x_2)$. It's clear then that if $\displaystyle P=\{a,x_1,x_2,b\}$ then $\displaystyle U\left(P,\mathbf{1}_c,\alpha\right)<\varepsilon$ from where integrability follows. The integral then follows by recalling that $\displaystyle \displaystyle \int_a^b\mathbf{1}_c\;d\alpha(x)=\sup_{P\in\wp}L(P ,\mathbf{1}_c,\alpha)=0$. The conclusion follows. $\displaystyle \blacksquare$


    From this note then that if $\displaystyle f\in\mathcal{R}(\alpha)$ on $\displaystyle [a,b]$ and $\displaystyle \tilde{f}$ is such that $\displaystyle f-\tilde{f}=\left(f(c)-\tilde{f}(c)\right)\mathbf{1}_c$ where $\displaystyle f(c)\ne\tilde{f}(c)$ then evidently $\displaystyle \tilde{f}$ is continuous since by our lemma we have that $\displaystyle \displaystyle \frac{1}{f(c)-\tilde{f}(c)}\left(f-\tilde{f}\right)$ is integrable and thus so is $\displaystyle \left(f(c)-\tilde{f}(c)\right)\cdot\frac{1}{f-\tilde{f}(c)}\left(f-\tilde{f}\right)f=\tilde{f}$ and


    $\displaystyle \displaystyle \int_a^b\tilde{f}(x)\text{ }d\alpha(x)=(f(c)-\tilde{f}(c))\int_a^b\mathbf{1}_c\text{ }d\alpha(x)+\int_a^bf(x)\text{ }d\alpha(x)=\int_a^bf(x)\text{ }d\alpha(x)$
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