Hello, @DanielFischer. I'm still having question regarding the existence of countably additive measure $\mu$ s.t. $\mu[a,b]=F(b)-F(a)$, as $F(x)$ is defined in the problem. I can prove the case when $\mu(a,b]=F(b)-F(a)$ (proof included in my problem description), but how about the case in my problem. Sorry to bother you again, but I really would like to get this problem resolved for sake of clarity. Thank you.

I guess you are a student at Courant and perhaps taking course with Varadhan?

Are you a Chinese student?

Let us add in WeChat?

Yeah, haha. I'm an undergraduate student taking the class. Sure, we can discuss problems in this class. I think this problem has some flaws, too.

yeah

are you Chinese?

since the proof is over [0,1], even if we use (a,b] the thing will be a little bit complicated. Since the total space is compact.

my WeChat is 9176800596

哈哈哈哈

我先去做饭了

那个HW2， 你可以看stein，下个pdf然后直接搜索dense这个词就能找到

证明基本是一样的