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Expand the logarithm of the product measure and exchange the finite sums:where part c identifies the expression in braces.

Use an independent copy $(x^{\prime },y^{\prime })$ of $(x,y)$ under the product measure $\Pi \otimes \Pi$. Write $q=y-x$, $q^{\prime }=y^{\prime }-x^{\prime }$, andSince the pushforward measure of $\Pi \otimes \Pi$ under $(z_t,z_t^{\prime })$ is ${\mu }_t\otimes {\mu }_t$, the interaction energy isTwo applications of the chain rule giveThe factor $1/2$ is retained. This classical identity requires differentiation under the integral sign to be justified, just as in part (b). Bounded Hessian matrix of $W$ is sufficient, since $|q-q^{\prime }{|}^2\le 2|q{|}^2+2|q^{\prime }{|}^2$ is integrable; compactly supported endpoints are another sufficient case. The printed quadratic-growth condition alone does not provide this domination.

The failure can occur for the interaction energy too. Take the even interaction potential $W(x)=1-\cos (2\pi x^3)$ and replace the endpoint probability measures in part (b) by $\frac{1}{2}{\delta }_0+\frac{1}{2}\mu$ and $\frac{1}{2}{\delta }_0+\frac{1}{2}\nu$. Keep the origin fixed and translate every other atom by one. This coupling is increasing on the line, hence is optimal: its assigned values extend to a nondecreasing transport map, which is the gradient of a convex function. Differences between translated atoms remain integers, so their mutual interaction vanishes. Only interactions with the origin contribute, giving $\mathcal{W}[{\mu }_t]=F(t)/4$, with $F$ from part (b). It therefore also lacks a finite classical second derivative at $t=1/2$.

Define the barycenter of a measure by $m_{\mu }=\int xd\mu (x)$. Under $\Pi$, the mean displacement is $\bar{q}=m_{\nu }-m_{\mu }$, and optimality gives $\int |q{|}^{2}d\Pi =D^2$. Expanding the square under the product measure yields the variance identity for independent copies:In the classical regime of part (c), the lower bound on the Hessian matrix of the interaction potential therefore givesFor equal barycenters, this becomes

For the full stated class of probability measures, prove the displacement convexity inequality directly. Pointwise strong convexity of $W$ givesIntegrating, including the $1/2$ in the interaction energy, and using the displayed variance identity for independent copies givesAll terms are finite by quadratic growth and finite second absolute moments. Equal barycenters give the claimed $\lambda$-displacement convexity. Also $m_{{\mu }_t}=(1-t)m_{\mu }+t{m}_{\nu }$, so this metric geodesic remains in the prescribed fixed-mean class. Applying the same argument to subintervals proves the second derivative bound in the sense of distributions even when a classical derivative has not been justified. The mean restriction is natural: translating all particles together leaves their pairwise differences and hence the interaction energy unchanged.

Consider the Borel setIts indicator function is nonnegative and measurable, so the Tonelli theorem applies to the product measure, without needing any separate absolute-integrability argument. At a fixed $y\in (a,b]$, its section in the first coordinate is $(a,y]$; outside $(a,b]$ that section is empty. ThereforeThe last equality uses the definition of a cumulative distribution function on the half-closed interval. It retains all point atoms of a measure on the diagonal $x=y$, excludes a possible point atom of a measure at $x=a$, and includes $y=b$, exactly as required by the set $D$. If $a=b$, both sides vanish. This verifies the identity even when the probability measures are not non-atomic measures.^[ref]

The product measure $\mu \otimes \nu$ is the unique probability measure on the sigma-algebra generated by measurable rectangles for whichHere the printed product notation $\mathcal{B}(\mathbb{R})\times \mathcal{B}(\mathbb{R})$ denotes the generated product sigma-algebra, conventionally written $\mathcal{B}(\mathbb{R})\otimes \mathcal{B}(\mathbb{R})$; it is not just the collection of rectangles. On ${\mathbb{R}}^2$ it equals $\mathcal{B}({\mathbb{R}}^2)$: coordinate projections show that rectangles are Borel sets, while open rectangles with rational endpoints give every open set as a countable union and hence generate the Borel sigma-algebra.

Existence is the product measure theorem for finite measures, which are in particular sigma-finite measures. Uniqueness follows because the rectangles form a pi-system generating the product sigma-algebra, and two finite measures agreeing there agree everywhere by the Dynkin lemma.^[ref]

For finitely many probability spaces $(X_i,{\mathcal{F}}_i,{\mu }_i)$, their product probability space has underlying set ${\prod }_i{X}_i$, the product sigma-algebra and product measure ${⨂}_i{\mu }_i$. The coordinate projections are jointly independent: for measurable $B_i\subseteq X_i$, the probability of all coordinate conditions is ${\prod }_i{\mu }_i(B_i)$. Conversely a family of independent coordinates has this product joint distribution. Product structure permits coordinate exposure and independent resampling in concentration arguments.

For a sequence of nontrivial increasing events on finite product measures, write $p_x$ for the parameter at which the success probability is $x$. An additive sharp threshold means that $p_{1-\epsilon }-p_{\epsilon }\to 0$ for every fixed $0<\epsilon <1/2$. When the threshold parameter itself tends to zero, relative sharpness instead compares this window with $p_{1/2}$; additive sharpness alone is then a weaker assertion. The Friedgut-Kalai sharp threshold theorem provides an explicit additive bound for transitive increasing events.

For two increasing events of equal probability $a$ under a product measure, positive association of their complements gives $1-P(A\cup B)\ge (1-a{)}^2$. Hence $a\ge 1-\sqrt{1-P(A\cup B)}$. For $r$ equal-probability increasing events, the same reasoning gives $a\ge 1-(1-P({\bigcup }_j{A}_j){)}^{1/r}$.

For an event $A$ in a product of finitely many probability spaces with its product measure, take the closed convex hull of the mismatch vectors $(1_{x_i\ne y_i}{)}_i$, $y\in A$. The distance of this hull from zero is $d_T(x,A)$. Equivalently it is the supremum, over nonnegative coordinate weights of squared sum at most one, of their minimum mismatch sum. Averaging mismatch patterns is the feature that distinguishes this quantity from ordinary Hamming distance.