= Solution
A real <Lévy process> starts at zero with probability one, has <stationary increments> and <independent increments>, is <stochastically continuous>, and is taken in its <càdlàg> version. Stationarity means $X_{t+s}-X_s$ has the law of $X_t$; independence means increments over disjoint ordered time intervals are independent. Stochastic continuity means $X_s\to X_t$ in probability as $s\to t$. One can equivalently impose starting at zero, stationary independent increments and stochastic continuity first, and then take a <càdlàg modification>.
A <Poisson random measure> with sigma-finite intensity $\nu$ on a measurable space $(E,\mathcal E)$ is a countably additive integer-valued random measure $N$ such that $N(A)$ has <Poisson distribution> with parameter $\nu(A)$ whenever $\nu(A)<\infty$, and counts on disjoint measurable sets are independent. A set of infinite intensity has infinite count with probability one. For jump processes the space is often time times a mark space, with intensity $dt\,\nu(dz)$. These conditions specify both the marginal count laws and their joint independence.
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