I am trying to show that, a set $E$ in $\left( 0,1\right) $ is such that, if $\left( \alpha,\beta\right) $ is any interval, then $\mu\left(E \cap \left( \alpha ,\beta \right) \right) \ge \delta \left( \beta -\alpha \right) $ where $\delta > 0 $ then the $\mu\left(E\right)=1$.
What have i tried. I could not help notice the case by case breakdown that $E$ might be completely contained in $\left( \alpha,\beta\right) $ in which case $\mu\left(E\right) = \mu\left(E \cap \left( \alpha ,\beta \right) \right) \ge \delta \left( \beta -\alpha \right)$
Similarly $E$ might have 2 parts so to calculate it's measure we'll need $\mu\left(E\right) = \mu\left(E \cap \left( \alpha ,\beta \right) \right) + \mu\left(E \backslash \left(E \cap \left( \alpha ,\beta \right) \right)\right)$
I am assuming a case of $\left(E \cap \left( \alpha ,\beta \right) \right) = \emptyset$ can not occur as that would imply $\alpha = \beta$ given the other conditions.
I was hoping if some one would be kind to give me a hint or a clue, so i could make progress.