GAN minimax objective and its distributional domain

Let $X$ be a set, let $\mathcal{A}$ be a sigma-algebra on $X$ (a family containing $\mathrm{\varnothing }$ and $X$, closed under complements and countable unions), and let $\mu :\mathcal{A}\to [0,+\mathrm{\infty }]$ be a countably additive measure with $\mu (\mathrm{\varnothing })=0$. Let $P,Q$ be probability measures on $(X,\mathcal{A})$, meaning countably additive measures with $P(X)=Q(X)=1$. A function from $X$ to a Borel space is measurable when the inverse image of each Borel set belongs to $\mathcal{A}$. Assume that $P$ and $Q$ have finite-valued, nonnegative, measurable densities $p$ and $q$ with respect to the same measure $\mu$, meaning that $P(E)={\int }_{E}pd\mu$ and $Q(E)={\int }_{E}qd\mu$ for every $E\in \mathcal{A}$. In particular, $\int pd\mu =\int qd\mu =1$.

For a measurable discriminator $D:X\to [0,1]$, put $a(t)=-\ln t$ for $0<t\le 1$ and $a(0)=+\mathrm{\infty }$, and put $b(t)=-\ln (1-t)$ for $0\le t<1$ and $b(1)=+\mathrm{\infty }$. In the products below, define $0\cdot (+\mathrm{\infty })=0$; a positive finite factor times $+\mathrm{\infty }$ is $+\mathrm{\infty }$. Define the log payoff and the unrestricted discriminator value by

$$V(D;P,Q):=-\int p(x)a(D(x))d\mu (x)-\int q(x)b(D(x))d\mu (x),C(P,Q):=\underset{D:X\to [0,1]\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}}{\sup }V(D;P,Q)\mathrm{.}$$

The functions $a$ and $b$ are Borel measurable: $\ln$ is continuous on $(0,\mathrm{\infty })$, and each extension assigns $+\mathrm{\infty }$ only at one Borel endpoint. Therefore $a\circ D$ and $b\circ D$ are measurable, and the products with $p$ and $q$ are measurable by the pointwise product convention and product-closure result referenced below. Thus both nonnegative integrals are defined.

Each integral is the nonnegative Lebesgue integral, defined as the supremum of the integrals of nonnegative simple minorants, so $V(D;P,Q)\in [-\mathrm{\infty },0]$. The supremum ranges over all measurable functions into $[0,1]$, including boundary values; it is not restricted to a finite-parameter discriminator family.

Measure-space convention

Sigma-algebra convention

Countably additive measure convention

Probability-measure convention

Nonnegative Lebesgue integral

Integral over a measurable set

Natural logarithm

Continuity of the natural logarithm

Measurable composition

Extended-real product convention

Measurability of arithmetic and products

Goodfellow et al., §§3–4.1, equations (1)–(3), complete 9-page paper

Li, §§1.1–1.2, equations (1)–(8), complete 6-page lecture note