Finite-sample likelihood ratio for iid observations

By [F8] with $g=1$, $\int \lambda d{Q}_0=1$, so $N=\{\lambda =\mathrm{\infty }\}$ is $Q_0$-null and hence $Q_1$-null. Set $\lambda =0$ on $N$; this preserves its density class and gives a finite nonnegative representative. Now fix $t\ge 1$ and $j\in \{0,1\}$. The vector $Y_t=(X_1,\dots ,X_t)$ is measurable into the finite product sigma-algebra because preimages of all-coordinate rectangles are finite intersections of coordinate events, and those rectangles generate that sigma-algebra by [F3]. Its law ${\mu }_{j,t}$ is a probability measure by [F5]. On every rectangle $R={\prod }_{i=1}^t{B}_i$, [F1] and [F2] give ${\mu }_{j,t}(R)={\prod }_i{Q}_j(B_i)$. Repeated application of [F4] constructs $Q_j^{\otimes t}$ with those same rectangle masses. The all-coordinate rectangles form a pi-system by [F3] and [F6], generate the product sigma-algebra, and have a finite exhaustion consisting of the whole space, whose mass is one for both measures. Hence [F6] gives ${\mu }_{j,t}=Q_j^{\otimes t}$.

Define ${\mathrm{\Lambda }}_t(x_1,\dots ,x_t)=(\lambda (x_1),\dots ,\lambda (x_t))$. Preimages of product rectangles are finite intersections of measurable coordinate preimages, so [F3] makes ${\mathrm{\Lambda }}_t$ measurable. The finite multiplication map $m_t:{\mathbb{R}}^t\to \mathbb{R}$, $m_t(y)={\prod }_{i=1}^t{y}_i$, is continuous: the two-coordinate multiplication map is continuous by $\mathrm{\mid }ab-a^{\mathrm{\prime }}{b}^{\mathrm{\prime }}\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b-b^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{b}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }a-a^{\mathrm{\prime }}\mathrm{\mid }$, and finite products follow by induction. Thus [F12] makes $m_t$ Borel and [F13] makes $h_t=m_t\circ {\mathrm{\Lambda }}_t$ measurable. By [F9], ${\nu }_t(B):={\int }_B{h}_{t}d{Q}_0^{\otimes t}$ is a measure. For each rectangle $R={\prod }_i{B}_i$, repeated Tonelli integration, justified by induction on the finite number $t$ of factors, gives ${\nu }_t(R)={\prod }_{i=1}^t{\int }_{B_i}\lambda d{Q}_0={\prod }_{i=1}^t{Q}_1(B_i)$, where each factor uses [F8]. This is the rectangle formula for $Q_1^{\otimes t}$ by [F4]. The two measures agree on the generating pi-system and both have total mass one, so [F6] gives ${\nu }_t=Q_1^{\otimes t}$.

If $A\in {\mathcal{F}}_t=\sigma (Y_t)$, then $A=Y_t^{-1}(B)$ for some product-measurable $B$: the family of such preimages is a sigma-algebra and is exactly the sigma-algebra generated by $Y_t$. The definition of law [F5] gives the integral change-of-variables identity first for indicators and nonnegative simple functions; increasing simple approximation and [F11] extend it to $h_t{\mathbf{1}}_B$. Thus, using steps 1.1 and 2.1, $P_1(A)=Q_1^{\otimes t}(B)={\int }_B{h}_{t}d{Q}_0^{\otimes t}={\int }_A{h}_t(Y_t)d{P}_0$. This is the Radon–Nikodym characterization of $L_t=h_t(Y_t)$.