Expectation = Probability?

Because no one else has replied, I'll post an answer (although it's been many years since I've dealt with limits of multiple integrals) ...

For convenience, I'm going to write $u,v,w$ in place of $xi,eta,zeta,$ so the quantity of interest is $$begin{align}&lim_{Delta x, Delta yto 0} {q(Delta x, Delta y)over Delta xDelta y}\ &= lim_{Delta x, Delta yto 0} {1over Delta xDelta y}int_{y_0}^{y_0+Delta y}duint_{0}^{M_2Delta x}dvint_{-M_2}^{v/Delta x}dw P(u,v,w)tag{1}\[2ex] &=lim_{Delta x, Delta yto 0} {1over Delta xDelta y}int_{y_0}^{y_0+Delta y}duint_{-M_2}^{0}dwint_{0}^{-w,Delta x}dv P(u,v,w)tag{2}\[2ex] &=lim_{Delta x, Delta yto 0} {1over Delta xDelta y}int_{y_0}^{y_0+Delta y}duint_{-M_2}^{0}dw P(u,c_{Delta x},w),(-wDelta x)tag{3}\[2ex] &=lim_{Delta x, Delta yto 0} {1over Delta xDelta y}int_{-M_2}^{0}dw P(c_{Delta y},c_{Delta x},w),(Delta y),(-wDelta x)tag{4}\[2ex] &=lim_{Delta x, Delta yto 0} int_{-M_2}^{0}dw P(c_{Delta y},c_{Delta x},w),(-w)tag{5}\[2ex] &=-int_{-M_2}^{0}dw P(y_0,0,w),wtag{6}\[2ex] end{align}$$

where

• $(1)implies(2)$ by changing the order of integration on $(v,w)$, noting that the region of integration in the $(v,w)$-plane is a triangle bounded by $0<v<M_2Delta x$ on the $v$-axis,$-M_2<w<0$ on the $w$-axis, and the line $w=-v/Delta x,$ i.e., the line $v=-wDelta x$.

• $(2)implies(3)$ by the mean value theorem for integrals, $(-wDelta x)$ being the width of the interval of integration on $v$, i.e. $[0,-wDelta x]$, and $c_{Delta x}$ being some value in that interval.

• $(3)implies(4)$, again by the mean value theorem, where $c_{Delta y}in[y_0,y_0+Delta y].$

• $(4)implies(5)$, dividing through by $Delta x,Delta y$.

• $(5)implies(6)$, because $c_{Delta y}$ and $c_{Delta x}$ are confined to intervals that converge to the points ${y_0}$ and ${0}$, respectively.

NB: This result is not obtained by considering it to be some sort of expectation value, even though the integrand is a product of $w$ and the joint probability density function evaluated at $(u,v)=(y_0,0).$ (An expectation requires the product to be $w$ and the marginal probability density for $w$, which is not obtained from the joint density by simply plugging in constants for $(u,v)$. Maybe it can be seen as the limit of some such?)