[proofplan] The density process is a conditional expectation process. Nonnegativity follows from the nonnegativity of the Radon-Nikodym derivative, integrability follows because $\mathbb Q$ is a probability measure, and the martingale property is exactly the [tower property of conditional expectation](/theorems/1150). The expectation identity follows from the averaging property and the fact that the total mass of $\mathbb Q$ is one. [/proofplan] [step:Verify nonnegativity, adaptedness, and integrability] Since $\mathbb Q\ll\mathbb P$, the Radon-Nikodym derivative \begin{align*} L&=\frac{d\mathbb Q}{d\mathbb P} \end{align*} is a nonnegative $\mathcal F$-measurable random variable. Since $\mathbb Q$ is a probability measure, \begin{align*} \mathbb E_{\mathbb P}[L] &= \int_{\Omega} L\,d\mathbb P \\ &= \mathbb Q(\Omega) \\ &=1. \end{align*} Thus $L$ is integrable under $\mathbb P$. For each $t$, the conditional expectation \begin{align*} Z_t&=\mathbb E_{\mathbb P}[L\mid\mathcal F_t] \end{align*} is $\mathcal F_t$-measurable and integrable. Moreover, conditional expectation preserves nonnegativity: since $L\geq0$ almost surely, $Z_t\geq0$ almost surely. [/step] [step:Apply the tower property to prove the martingale identity] Let $0\leq s\leq t\leq T$. Since $(\mathcal F_t)_{0\leq t\leq T}$ is a filtration, $\mathcal F_s\subseteq\mathcal F_t$. By the [Tower Property of Conditional Expectation](/theorems/1150), \begin{align*} \mathbb E_{\mathbb P}[Z_t\mid\mathcal F_s] &= \mathbb E_{\mathbb P}[\mathbb E_{\mathbb P}[L\mid\mathcal F_t]\mid\mathcal F_s] \\ &= \mathbb E_{\mathbb P}[L\mid\mathcal F_s] \\ &= Z_s \end{align*} almost surely. Hence $(Z_t)_{0\leq t\leq T}$ is a martingale under $\mathbb P$. [/step] [step:Compute the expectation of the density process] Using the averaging property of conditional expectation, \begin{align*} \mathbb E_{\mathbb P}[Z_t] &= \mathbb E_{\mathbb P}[\mathbb E_{\mathbb P}[L\mid\mathcal F_t]] \\ &= \mathbb E_{\mathbb P}[L] \\ &=1. \end{align*} This proves $\mathbb E_{\mathbb P}[Z_t]=1$ for every $0\leq t\leq T$. [/step]