Consider the measurable space $(\mathbb{R}, \mathcal{B}(\mathbb{R})$, $\lambda)$, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra of $\mathbb{R}$ and $\lambda$ is the Lebesgue measure restricted to it. For $A \in \mathcal{B}(\mathbb{R})$, let $$ \mu(A):=\int_A e^{-x} d \lambda(x) . $$ My question is about how to compute the measure of the set (A) when it takes the form of intervals, such as $\mu([0, \infty])$ or $\mu(\mathbb{R})$, where in the first case there are only positive values and in the second, both positive and negative values.
I'm trying to apply the Lebesgue measure definition to this specific case: \begin{equation*} \lambda(A):=\operatorname{inf}\left\{\sum_{n=1}^{\infty} v\left(I_n\right) ; A \subseteq \bigcup_{n \in \mathbb{N}} I_n: n \in \mathbb{N}, I_n \in \mathcal{I}^N,\right\} \end{equation*} where $\mathcal{I}^N$ is the family of bounded intervals, and $v(I_n)$ represents the lengths of the intervals, but i can't get any satisfactory reuslts. In my example, it's one dimension, hence $n=1$.
The integration is with respect to the variable $\lambda(x)$ over the chosen set $A$, when it takes the form $[0, \infty]$ or $\mathbb{R}$.
First Edition:
I think a possible solution is to take intervals as follows and multiply it by exponential $$ \begin{gathered} \lambda([b, \infty))=\lambda([b, b+1) \cup[b+1, b+2) \ldots)=\lambda\left(\bigcup_{k=0}^{\infty}[b+k, b+k+1)\right)= \\ =\sum_{k=0}^{\infty} \lambda([b+k, b+k+1))=\sum_{k=0}^{\infty} 1=\infty \end{gathered} $$ And the last, multiply by the exponential \begin{equation} -e^{-x}\cdot\lambda([b, \infty))= -e^{-x}\cdot\lambda([0, \infty))=-e^{-x}\cdot\infty \end{equation} But I’m not sure if the interval also applies to the exponential or not. I think so