I know there are lots of questions like this question, but I think the question I have is pretty basic, and I could imagine this formulation is useful. Suppose $G$ is a distribution function with bounded support–denote an upper bound by $K$.
Let $dG$ denote the Lebesgue-Stieltjes measure wrt to $G$ and let $dy$ denote the L-S measure wrt the identity, (i.e. the Lebesgue measure). Can I say that:
$$\int_{(x, K]}ydG + \int_{(x, K]} G dy = K G(K) – x G(x).$$
Is this valid? Do I need a left limit on G(x)? I suppose the question boils down to the "right" integration-by-parts formula for Lebesgue-Stieltjes integrals. What I see here suggests I need a left limit on G, but I have defined the interval of integration slightly differently. This made me wonder if I do not need any limits, since the identity sort of stands in for "$F$" and is continuous. But that question and the answers don't really provide references or derivations that would make one confident.
Edit: If I can use a version of this formula, but need a left limit here, would I be able to integrate on $(-K,x]$ without the left limit term?
Best Answer
Yes, your formula is correct. Of the four intervals $\ (a,b), (a,b], [a,b)\ $ and $\ [a,b]\ $, $\ (a,b]\ $ is the only one for which no limits need to be taken at either end. The formulas for the probabilities that a random variable $\ X\ $ lies in various intervals, in terms of its distribution function $\ F_X\ $, are good crutches for remembering the rules about what limits are needed. For $\ b>a\ $, \begin{align} \mathbb{P}(X\in(a,b)\,)&=\lim_{x\rightarrow b^-}F_X(x)-F_X(a)\\ \mathbb{P}(X\in(a,b]\,)&=F_X(b)-F_X(a)\\ \mathbb{P}(X\in[a,b)\,)&=\lim_{x\rightarrow b^-}F_X(x)-\lim_{x\rightarrow a^-}F_X(x)\\ \mathbb{P}(X\in[a,b]\,)&=F_X(b)-\lim_{x\rightarrow a^-}F_X(x)\ . \end{align} All of these rules except the first also hold for $\ b=a\ $.
So, also, for your integration by parts, \begin{align} \int_{(x, K)}ydG + \int_{(x, K)} G dy &= K \lim_{y\rightarrow K^-}G(y) - x G(x)\\ \int_{(x, K]}ydG + \int_{(x, K]} G dy &= K G(K) - x G(x)\\ \int_{[x, K)}ydG + \int_{[x, K)} G dy &= K \lim_{y\rightarrow K^-}G(y) - x \lim_{y\rightarrow x^-}G(y)\\ \int_{[x, K]}ydG + \int_{[x, K]} G dy &= K G( K) - x \lim_{y\rightarrow x^-}G(y)\ , \end{align} with the same conditions on $\ x\ $ and $\ K\ $ as on $\ a\ $ and $\ b\ $ above.