I have a question about the measurability of a certain set. Let $(\mathbb{X}, \mathcal{X})$ be a measurable space. Let $$\mathbf{N} := \left \{ \sum_{n = 0}^N \delta_{x_k} : N \in \mathbb{N}_0 \cup \{\infty \}, x_k \in \mathbb{X} \right \}$$ where $\delta_{x_k}$ denotes the Dirac-measure. We endow $\mathbf{N}$ with the smallest $\sigma$-algebra such that $\mu \mapsto \mu(A)$ is measurable for all $A \in \mathcal{X}$, denoted by $\mathcal{N}$. For example, these notations are used in https://www.math.kit.edu/stoch/~last/seite/lectures_on_the_poisson_process/media/lastpenrose2017.pdf, page 10 to introduce Poisson processes on general spaces.
Is the set $$B := \left \{ \mu \in \mathbf{N} : \mu(\{x\}) < \infty \text{ for all } x \in \mathbb{X} \right \}$$ contained in $\mathcal{N}$?
Intuitively, I would say that this set is non-measurable for uncountable $\mathbb{X}$, however I find it hard to show non-measurability.