1. ## Measurable Subset Question

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

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

Is this correct?

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

I suppose the answer is 1?

2. ## Re: Measurable Subset Question

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

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