# Thread: real analysis...pls help me with this!

1. ## real analysis...pls help me with this!

Let E be a nonempty subset of R. Find an open set U of R that satisfies the following two conditions: E C U and m*(U) ≤m*(E) + ¼ [20 marks]

2. Originally Posted by cynthiacheok
Let E be a nonempty subset of R. Find an open set U of R that satisfies the following two conditions: E C U and m*(U) ≤m*(E) + ¼ [20 marks]
The usual definition of outer measure m* is: $m^*(E) = \inf\left\{\sum m(U_n)\right\}$, where the infimum is taken over all countable collections of open intervals $U_n$ such that $E\subseteq\bigcup U_n$.

That means that we can find such a countable collection satisfying $m^*(E) \leqslant \sum m(U_n) + \textstyle\frac14$. Define $U = \bigcup U_n$. Then E ⊆ U. Also, $m(U)\leqslant\sum m(U_n)$ and so $m(E) \leqslant m(U) + \textstyle\frac14$.