Under which conditions is a set that contains only null sets a null set as well?

Question

Let $(X_t)_t$ be an a.s. cadlag process defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and the measure space $(E,\mathcal{E})$ be its state space. Now we define $$ X:\Omega\to E^{\mathbb{R_+}} $$ by $X(\omega)=(t\mapsto X_t(\omega))$. Furthermore $\mathcal{E}^{\mathbb{R}_+}$ be the smallest $\sigma$-algebra on $E^{\mathbb{R}_+}$ that contains all cylinder sets $\{x\in E^{\mathbb{R}_+}:x(t)\in A\}$ for $t\geq 0,A\in\mathcal{E}$. Furthermore $\mathcal{P}^X$ be the law of $X$. What I want now is that $\mathbb{P}^X$-almost every $x\in E^{\mathbb{R}_+}$ is cadlag as well. Unfortunatetly the image of $X$ ($Im(X)$) doesn't have to be measurable. But we see that every measurable set that contains $Im(X)$ has measure $1$. Or equivalent every measurable set contained in $Im(X)^c$ has measure zero. Can I use this some how to prove that $Im(X)^c$ is a null set?

Answer 1

Example: Take $\Omega$ to be the space $C([0,1]\to\Bbb R)$ (viewed as a metric space under the uniform-distance metric) equipped with its Borel $\sigma$-field $\mathcal F$ and Wiener measure $\Bbb P$. Take $X_t(\omega)=\omega(t)$ for $t\ge 0$ and $\omega\in\Omega$. You can check that the $\Bbb P^X$ measure of any element of ${\mathcal E}^{\Bbb R_+}$ contained in $Im(X)$ is zero.