Equivalence Definitions for Nonhomogeneous Poisson Process By Stocastic Processes, Sheldon M. Ross, The Second Edition, p.78, the definition of nonhomogeneous Poisson process is given by:

The counting process $\{N(t),t\geq 0\}$ is said to be a nonstationary or nonhomogeneous Poisson process with intensity function $\lambda(t),t\geq 0$ if
(i) $N(0) = 0$
(ii) $\{N(t),t\geq 0\}$ has independent increments
(iii) $P\{N(t+h) - N(t)\geq 2\}=o(h)$
(iv) $P\{N(t+h)-N(t) = 1\}=\lambda(t)h+o(h)$

Here $o(h)$ is a function of $h$ such that
$$
\lim_{h\to 0}{\frac{o(h)}{h}} = 0.
$$
On p. 79, he said

If we let
$$
m(t) = \int_0^t{\lambda(s)\mathrm{d}s},
$$
then it can be shown that
\begin{equation}
P\{N(t+s)-N(t) = n\} = \exp\{-(m(t+s)-m(t))\}[m(t+s)-m(t)]^n/n!,\quad n\geq 0\tag{$*$}
\end{equation}
That is, $N(t+s)-N(t)$ is Poisson distributed with mean $m(t+s)-m(t)$.

My question is that, does there exist an equivalence definition for nonhomogeneous Poisson process, just like the homogeneous one, like
"The counting process $\{N(t),t\geq 0\}$ is said to be a nonhomogeneous Poisson process if
(i) $N(0)=0$
(ii) The process has independent increments
(iii) There exists a function $m(\cdot)$ such that $(*)$ is true for all $s,t\geq 0$"?
If so, what will be the intensity function $\lambda(t)$? Will it be
$$
\lambda(t) = \frac{\mathrm{d}}{\mathrm{d}t}m(t)?
$$
 A: In the book: K. Itô, Essentials of Stochastic Processes, American Mathematical Society(Translation of Mathematical Monographs; v.231), Providence, RI.(2006)  Th 2.9.2-3, p.33-35, there are following theorems:
Theorem 2.9.2. If $X_t, t\in T=[a,b) $ is a separable Poisson process in wide sense,  then with probability 1, its sample process is a step function which increases only by jumps of magnitude 1.
Theorem 2.9.3. If an additive process which is continuous in probability has the property that, with probability 1, its sample process is a step function
which increasing only with jumps of magnitude 1, then it is a separable Poisson process in a wide sense.
In Ito's book, the additive process $X_t$-- process with independent increments and $X_0=0$.
Poisson process in a wide sense -- a process $X_t$ with independent increments, which is continuous in probability and the distribution of $X_t-X_s$ for each pair $t>s$ is a Poisson distribution.
From above theorems, if $N=\{N_t, t\ge 0\}$ is a stochastic continuous process, then
the $N$ is Poisson process in a wide sense(i.e., nonhomogeneous Poisson process) may be
characteristicrized as $N$ is a simple counting process with independent increments,
where simple counting process means that its sample process is a step function
which increasing only with jumps of magnitude 1.
Now, if $N$ is a nonhomogeneous Poisson process, then $N_t-N_s$ is a Poisson distributed r.v. for each pair $t>s$ and
\begin{equation*}
 m(t)=\mathsf{E}[N_t] \tag{1}
\end{equation*}
is a non-negative increasing continuous function of $t$. Furthermore, the $N_t-N_s$ is
Poisson distributed as
\begin{equation*}
\mathsf{P}(N_{t+s}-N_t=k)=\frac{[m(t+s)-m(t)]^k}{k!}e^{-[m(t+s)-m(t)]}, \qquad k=0,1,2,\cdots.
\end{equation*}
If $m(t)$ in (1) is an absolutely continuous function, or epuivalently, the L-S measure $\mathrm{d}m$ generated by $m$ is absolute continuous with respect to the Lebesgue measure $\mathrm{d}t$,
and $\lambda(t)=\frac{dm}{dt}(t)$, then
\begin{equation}
 m(t)=\int_{0}^{t}\lambda(s)\,\mathrm{d}s.
\end{equation}
