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Thread: Measurable Subset Question

  1. #1
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    Measurable Subset Question

    1)Let $\displaystyle E$ be a measurable subset of $\displaystyle \mathbb{R}$ with $\displaystyle m(E)=1$ and let $\displaystyle (E_n)$ be a sequence of measurable subsets of $\displaystyle E$. If, for any $\displaystyle \epsilon > 0$, there exists a set $\displaystyle E_i$ in the sequence $\displaystyle (E_n)$ with $\displaystyle m(E_i) > 1 - \epsilon$, find the value of $\displaystyle m(\cup E_n)$.

    My working is as below:
    Since $\displaystyle m(E) = 1$ & $\displaystyle m(E_i) > 1 - \epsilon$,
    $\displaystyle m(E_i) > m(E) - \epsilon$, so $\displaystyle m(E \setminus E_i) < \epsilon$.
    So set $\displaystyle \epsilon = \frac{1}{n}$ such that $\displaystyle m(E \setminus E_n) < \epsilon$ for $\displaystyle n \geq i$
    So $\displaystyle m(E \setminus \cup E_n) = 0$.
    Hence $\displaystyle m(\cup E_n) = 1$.

    Is this correct?

    2)Let $\displaystyle E$ be a measurable set with $\displaystyle m(E) = 1$. Suppose that $\displaystyle {E_n}$ is a sequence of measurable subsets of $\displaystyle E$ such that $\displaystyle m(E_n) = 1$ for $\displaystyle n=1,2,3,...$. Find $\displaystyle m(\cap E_n)$.

    I suppose the answer is 1?
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  2. #2
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    Re: Measurable Subset Question

    Quote Originally Posted by Markeur View Post
    2)Let $\displaystyle E$ be a measurable set with $\displaystyle m(E) = 1$. Suppose that $\displaystyle {E_n}$ is a sequence of measurable subsets of $\displaystyle E$ such that $\displaystyle m(E_n) = 1$ for $\displaystyle n=1,2,3,...$. Find $\displaystyle m(\cap E_n)$.

    I suppose the answer is 1?
    Yes, because each of the complementary sets $\displaystyle E\setminus E_n$ is a null set, and the union of countably many null sets is null.
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