# Integration of subdifferential.

For example, for the function $$f(x)=|x|,\quad\text{-1\le x\le1,}$$ the subdifferential $D^{+}f(x)$ is $$D^{+}f(x)=\begin{cases} -1\quad&\text{for x\in[-1,0),}\\ [-1,1]\quad&\text{at x=0,}\\ 1\quad&\text{for x\in(0,1].} \end{cases}$$

I now want to know what the the following integral is: (formally) $$\int_{-1}^{1}D^{+}f(x)dx=?.$$ However, as you see, $D^{+}f$ is multi-valued function. I thought that it is alright to interpret this integral as Lebersgue integral since we may ignore the point $x=0$, that is, $$\int_{-1}^{1}D^{+}f(x)dx=-1\cdot|[-1,0)|+1\cdot|(0,1]|=0,$$ but what happend if we interpret in the sense of Riemann integral? Can we interpret in this sense in the first place? I have searched but I didn't find any information.

Please give me some comments for my question and my interpretation in the sense of Lebesgue integral if you know.

Let $f:I\to\mathbb{R}$ be a continuous convex function and let $\varphi:I\to\mathbb{R}$ be a function such that $\varphi(x)\in\partial f(x)$ for every $x\in\operatorname{int} I$. Then for every $a<b$ in $I$ we have $$f(b)-f(a)=\int^b_a\varphi(t)dt.$$
• Does such a result where $I$ is replaced by a Hilbert space? – AIM_BLB Feb 4 at 7:06