Suppose that $W(t)$ is a Brownian Motion, $N(ds,dz)$ is a Possion random measure with intensity $ds\mu(dz)$,$\tilde{N}(ds,dz)$ is the compensated measure. Shall we call $$X(t)=W(t)+\int_0^t\int_0^{1}(e^z-1)\tilde{N}(ds,dz)+\int_0^t\int_1^{\infty}(e^z-1)N(ds,dz)$$ a Levy-ito decomposition?
If $M(ds,dz)$ is another Possion measure with intensity $ds\nu(dz)$ on the same Probability space, and $$X(t)=W(t)+\int_0^t\int_0^{1}z\tilde{M}(ds,dz)+\int_0^t\int_1^{\infty}(z)M(ds,dz),$$
then what is the relationship of $N(ds,dz)$ and $M(ds,dz)$?