0

Suppose $f \in BV_{loc}(\mathbb{R})$ and $g \in C^{1}(\mathbb{R})$. Also assume that $f$ has compact support (so I guess $f \in BV(\mathbb{R})$ in fact).

Then do we have the integration by parts formula:

$$\int_{\mathbb{R}} f(x)g'(x)~dx = - \int_{\mathbb{R}} g(x)f'(x)~dx $$ if $f'$ is understood as a measure? I know that since $f \in BV_{loc}$ then its distributional derivative is a signed Radon measure. I only know that this formula holds for absolutely continuous functions, I'm not sure about a weaker case like this.

duelspace
  • 1,235

1 Answers1

1

I can give you three possible cases of such problems.

Let $f,g:[a,b]\rightarrow \mathbb{R}$ be Lipschitz continuous.

Let $f:[a,b]\rightarrow \mathbb{R}$ be absolutely continuous function, and let $g:[a,b]\rightarrow \mathbb{R}$ be a continuously differentiable function.

Let $f,g:\mathbb{R} \rightarrow \mathbb{R}$ be monotone non-decreasing and continuous functions, when we need to consider in Stieltjes sense.

Mangoo
  • 98
  • Do you mind sharing more about the third scenario on why the integration by parts formula hold? As the first two scenarios are just special cases when $f$ and $g$ are both absolutely continuous. – L-JS Jun 30 '23 at 18:15
  • You can find it in "An introduction to measure theory" by Terence Tao, it is exercise 1.7.16. – Mangoo Jul 01 '23 at 06:19