Lebesgue outer measure of the union of a countable sequence of disjoint intervals in R is the sum of respective length of each intervals

Question

What I want to prove is that if $\left \{ I_{n} \right \}$ is a sequence of disjoint open sets, then we have $L\left ( \bigcup_{k= 1}^{\infty }I_{k} \right )= \sum_{k= 1}^{\infty }L\left ( I_{k} \right )$ , here L(A) represent the lebesgue outer measure of A, I can only prove the direction of $L\left ( \bigcup_{k= 1}^{\infty }I_{k} \right )\leq \sum_{k= 1}^{\infty }L\left ( I_{k} \right )$ via the countable subadditivity of lebesgue outer measure, now I want to know how to prove the another direction? and I also wonder whether this theorem still holds if the condition of this problem is changed to "$\left \{ I_{n} \right \}$ is a sequence of intervals and any two of these intervals only intersect with each other at the end points"I think it is true because the set of all the end points is countable and countable set has lebesgue outer measure 0.