Poisson process vs. Poisson Random Measure

Question

Let $(W_i)_{i \in \mathbb{N}}$ be iid Exp($\lambda$) and $T_n := \sum_{i=1}^n W_i$. Then it is well-known that $$ N(t) := \sum_{n \in \mathbb{N}} 1_{T_n \leq t} = \sum_{n \in \mathbb{N}} \varepsilon_{T_n}([0,t])$$

defines a homogenous Poisson process with intensity $\lambda$, i.e.

1. $N(0) = 0$
2. $N(t) - N(s) \sim Poi(\lambda (t-s))$ for all $0 \leq s \leq t$
3. For $0 \leq t_0 < t_1 < \dots < t_n$ the increments $N(t_i)-N(t_{i-1})$, $i=1, \dots, n$ are independent.

How can one then show that the very similar process $$\tilde{N}(A) := \sum_{n \in \mathbb{N}} \varepsilon_{T_n}(A), \hspace{1cm} A \in \mathcal{B}([0, \infty))$$ is a Poisson random measure with mean measure $\mu = \lambda \vert \cdot \vert$, where $\vert \cdot \vert$ denotes the Lebesgue measure$, i.e. how can one show that

1. $\tilde{N}(A) \sim Poi(\mu(A))$ for all $A \in \mathcal{B}([0, \infty))$
2. $\tilde{N}(A_1), \dots, \tilde{N}(A_n)$ are independent for mutually disjoint $A_i \in \mathcal{B}([0, \infty))$

Certainly both properties hold for intervals, but how can one conclude that they also hold for general Borel sets?