Dependencies

For \(\alpha \in (0,1)\), if \(\operatorname{eR}_\alpha (\mathcal{P}, \mathcal{Q}) {\lt} \infty \) and there exists a measurable set \(A\) with \(P(A^c) = 0\) and \(Q(A^c) = 0\) for all \(P \in \mathcal{P}\) and \(Q \in \mathcal{Q}\), then

That is, the infimum defining \(\operatorname{eR}_\alpha (\mathcal{P}, \mathcal{Q})\) can be restricted to measures \(R\) supported on \(A\).

Let \(P \in \mathcal{M}(\mathcal{X})\) and let \(\mathcal{Q} \subseteq \mathcal{M}(\mathcal{X})\) be a set of measures. We say that \(P\) is absolutely continuous with respect to \(\mathcal{Q}\) (denoted by \(P \ll \mathcal{Q}\)) if all \(\mathcal{Q}\)-null sets are \(P\)-null sets.

Let \(\mathcal{Q} \subseteq \mathcal{M}(\mathcal{X})\) be a set of measures. A set \(S \subseteq \mathcal{X}\) is said to be \(\mathcal{Q}\)-null if \(Q[S] = 0\) for all \(Q \in \mathcal{Q}\). A property \(p : \mathcal{X} \to \mathrm{Bool}\) is said to hold \(\mathcal{Q}\)-almost everywhere (denoted by \(\mathcal{Q}\)-a.e.) if the set \(\{ x \mid \neg p(x)\} \) is \(\mathcal{Q}\)-null.

A jointly measurable function \(f : \mathcal{X} \times \mathcal{Y} \to \mathbb {R}_{+, \infty }\) is a conditional e-variable for a set of Markov kernels \(H \subseteq (\mathcal{X} \rightsquigarrow \mathcal{Y})\) if \(f(x, \cdot )\) is in \(\mathcal{E}_{H(x)}\) for all \(x \in \mathcal{X}\) (in which \(H(x) = \{ \eta (x) \mid \eta \in H\} \)). We denote the set of conditional e-variables for \(H\) by \(\mathcal{E}_H\).

Let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be a Markov kernel and \(H \subseteq (\mathcal{X} \rightsquigarrow \mathcal{Y})\) a set of Markov kernels. A conditional e-variable \(f\) for \(H\) is a conditional numeraire for \((\kappa , H)\) if for all \(x \in \mathcal{X}\), the function \(f(x, \cdot )\) is a numeraire for \((\kappa (x), H(x))\). We denote the conditional numeraire for \((\kappa , H)\) by \(E^*_{\kappa , H}\) (if it exists).

For a set of probability measures \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\), the effective set of measures \(\mathcal{Q}_{\operatorname{eff}}\) is the set \(\{ \mu \in \mathcal{M}_+(\mathcal{X}) \mid \forall f \in \mathcal{E}_{\mathcal{Q}}, \: \mu [f] \le 1\} \).

For \(f : \Omega \to \overline{\mathbb {R}}\) and \(\mu \) a measure on \(\Omega \), we define the extended integral of \(f\) with respect to \(\mu \) as

where \(f^{+} = \max (f, 0)\) and \(f^{-} = -\min (f, 0)\), and their integrals are Lebesgue integrals, possibly taking the value \(+\infty \).

The set of extended non-negative real numbers is \(\mathbb {R}_{+,\infty } = \mathbb {R}_{+} \cup \{ +\infty \} \).

The logarithm as a function from \(\mathbb {R}_{+, \infty }\) to \(\overline{\mathbb {R}}\) is defined as

The set of extended real numbers is \(\overline{\mathbb {R}} = \mathbb {R} \cup \{ +\infty , -\infty \} \).

The e-Rényi divergence of order \(\alpha \in (0,1)\) between two sets of probability measures \(\mathcal{P}, \mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) is defined as

for \(\mathcal{U}_{\log }\) the max-utility corresponding to the logarithmic utility.

An E-variable for a set of probability measures \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) is a measurable function \(f : \mathcal{X} \to \mathbb {R}_{+,\infty }\) such that for all \(Q \in \mathcal{Q}\), \(Q[f] \le 1\).

Let \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). A numeraire for \((P, \mathcal{Q})\) is a \(P\)-almost surely positive e-variable \(f^* \in \mathcal{E}_{\mathcal{Q}}\) such that \(P[f / f^*] \le 1\) for all \(f \in \mathcal{E}_{\mathcal{Q}}\) .

The logarithm satisfies the properties of a utility function and we call it logarithmic utility.

For a utility function \(U\), the maximal randomized utility for \((P, \mathcal{Q})\) is defined as

in which the expectation is the extended integral defined in Definition 1.4.

For a utility function \(U\), the maximal utility for \((P, \mathcal{Q})\) is defined as

in which the expectation is the extended integral defined in Definition 1.4.

A randomized E-variable for a set of distributions \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) is a Markov kernel \(\eta : \mathcal{X} \rightsquigarrow \mathbb {R}_{+,\infty }\) such that for all \(Q \in \mathcal{Q}\), \((\eta \circ Q)[\operatorname{id}] \le 1\).

Let \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). The reverse information projection (RIPr) of \(P\) onto \(\mathcal{Q}\) is the finite measure \(P^*_{\mathcal{Q}}\) absolutely continuous with respect to \(P\) such that

It satisfies \(P^*_{\mathcal{Q}}[\mathcal{X}] \le 1\) and the inequality can be strict.

The singular set of \(P\) with respect to \(\mathcal{Q}\) is a \(\mathcal{Q}\)-null set \(A\) such that the singular part \(P_{\perp }\) satisfies \(P_{\perp }(A^c) = 0\).

We call utility a function \(U : \mathbb {R}_{+,\infty } \to \overline{\mathbb {R}}\) which is non-decreasing, concave, finite except perhaps at 0 and \(\infty \), continuous, and continuously differentiable on \((0, \infty )\).

Let \(P \in \mathcal{M}(\mathcal{X})\) and let \(\mathcal{Q} \subseteq \mathcal{M}(\mathcal{X})\). Then \(P \ll \mathcal{Q}\) iff \(\operatorname{ae}_P \le \bigsqcup _{Q \in \mathcal{Q}} \operatorname{ae}_Q\).

A set \(S \subseteq \mathcal{X}\) has a \(\mathcal{Q}\)-null complement iff \(S \in \bigsqcup _{Q \in \mathcal{Q}} \operatorname{ae}_Q\).

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and \(U : \mathbb {R}_{+,\infty } \to \overline{\mathbb {R}}\) a non-decreasing concave \(C^1\) function which is bounded from above. Then the e-variable \(f^*\) in Lemma 2.36 is such that for all e-variables \(g \in \mathcal{E}_{\mathcal{Q}}\),

If there exists a conditional numeraire for \((\kappa , \{ \eta \} )\), then \(1/E^*_{\kappa , \{ \eta \} }\) is a Radon-Nikodym derivative for \(\eta \) with respect to \(\kappa \).

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures. The effective set \(\mathcal{Q}_{\operatorname{eff}}\) is convex and satisfies \(0 \in \mathcal{Q}_{\operatorname{eff}}\) and \(\mathcal{Q} \subseteq \mathcal{Q}_{\operatorname{eff}}\).

The function \(R \mapsto \mathcal{U}(R, \mathcal{P})\) is convex.

Let \(\mathcal{P}' \subseteq \mathcal{P}\) and \(\mathcal{Q}' \subseteq \mathcal{Q}\). Then

Let \(\delta \in [0,1/2]\). Let \(B_{\le \delta } = \{ \operatorname{Ber}(p) \mid p \le \delta \} \) and \(B_{\ge 1 - \delta } = \{ \operatorname{Ber}(p) \mid p \ge 1 - \delta \} \) be sets of Bernoulli distributions with mean constraints. Then

Let \(\delta \in [0,1/2]\). Let \(\mathcal{D}_{\le \delta } = \{ P \in \mathcal{P}([0,1]) \mid P[\operatorname{id}] \le \delta \} \) and \(\mathcal{D}_{\ge 1 - \delta } = \{ P \in \mathcal{P}([0,1]) \mid P[\operatorname{id}] \ge 1 - \delta \} \) be sets of distributions on \([0,1]\) with mean constraints. Then

Let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be Markov kernel. Then

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a measurable map. Then

Let \(\mathcal{P},\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be two sets of probability measures. Suppose that there exists a measurable map \(\phi : \mathcal{X} \to \mathcal{X}\) such that \(\phi \circ \phi = \operatorname{id}\) and \(\phi _* \mathcal{P} = \mathcal{Q}\). Then

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) and let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be a Markov kernel. Then \(\kappa \circ \mathcal{Q}_{\operatorname{eff}} \subseteq (\kappa \circ \mathcal{Q})_{\operatorname{eff}}\).

Let \(\mathcal{Q}_1 \subseteq \mathcal{Q}_2 \subseteq \mathcal{P}(\mathcal{X})\) be two sets of probability measures. Then \(\mathcal{Q}_{1, \operatorname{eff}} \subseteq \mathcal{Q}_{2, \operatorname{eff}}\).

Let \(Q \in \mathcal{P}(\mathcal{X})\) be a probability measure. Then \(\{ Q\} _{\operatorname{eff}} = \{ Q' \in \mathcal{M}(\mathcal{X}) \mid Q' \le Q\} \), in which \(Q' \le Q\) means that \(Q'(A) \le Q(A)\) for all measurable sets \(A \subseteq \mathcal{X}\).

Let \(\mathcal{P}' \subseteq \mathcal{P}\) and \(\mathcal{Q}' \subseteq \mathcal{Q}\). Then, for all \(\alpha \in (0, 1)\),

Let \(\delta \in [0,1/2]\). Let \(B_{\le \delta } = \{ \operatorname{Ber}(p) \mid p \le \delta \} \) and \(B_{\ge 1 - \delta } = \{ \operatorname{Ber}(p) \mid p \ge 1 - \delta \} \) be sets of Bernoulli distributions with mean constraints. Then

Let \(\delta \in [0,1/2]\). Let \(\mathcal{D}_{\le \delta } = \{ P \in \mathcal{P}([0,1]) \mid P[\operatorname{id}] \le \delta \} \) and \(\mathcal{D}_{\ge 1 - \delta } = \{ P \in \mathcal{P}([0,1]) \mid P[\operatorname{id}] \ge 1 - \delta \} \) be sets of distributions on \([0,1]\) with mean constraints. Then

Let \(a \le b \in [0,1]\). Let \(\mathcal{D}_{\le a} = \{ P \in \mathcal{P}([0,1]) \mid P[\operatorname{id}] \le a\} \) and \(\mathcal{D}_{\ge b} = \{ P \in \mathcal{P}([0,1]) \mid P[\operatorname{id}] \ge b\} \) be sets of distributions on \([0,1]\) with mean constraints. Let \(B_{\le a} = \{ \operatorname{Ber}(p) \mid p \le a\} \) and \(B_{\ge a} = \{ \operatorname{Ber}(p) \mid p \ge a\} \) be sets of Bernoulli distributions with the same constraints. Then

Let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be Markov kernel. Then

For \(\alpha \in (0,1)\), if \(\operatorname{eR}_\alpha (\mathcal{P}, \mathcal{Q}) {\lt} \infty \), then

That is, the infimum defining \(\operatorname{eR}_\alpha (\mathcal{P}, \mathcal{Q})\) can be restricted to measures \(R\) that are absolutely continuous with respect to both \(\mathcal{P}\) and \(\mathcal{Q}\).

TODO: can we drop the finiteness assumption?

Let \(\mathcal{P},\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be two sets of probability measures. Suppose that there exists a measurable map \(\phi : \mathcal{X} \to \mathcal{X}\) such that \(\phi \circ \phi = \operatorname{id}\) and \(\phi _* \mathcal{P} = \mathcal{Q}\). Then

Let \(\phi : \mathcal{X} \to \mathcal{Y}\) be a measurable embedding. Then

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a measurable map. Then

Let \(\mathcal{P},\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be two sets of probability measures. Suppose that there exists a measurable map \(\phi : \mathcal{X} \to \mathcal{X}\) such that \(\phi \circ \phi = \operatorname{id}\) and \(\phi _* \mathcal{P} = \mathcal{Q}\). Then

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and let \(\phi : \mathcal{X} \to \mathcal{Y}\) be a measurable function. Then for all \(g \in \mathcal{E}_{\phi _* \mathcal{Q}}\), \(g \circ \phi \in \mathcal{E}_{\mathcal{Q}}\).

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) and \(H \subseteq (\mathcal{X} \rightsquigarrow \mathcal{Y})\) be a set of Markov kernels. If \(f \in \mathcal{E}_{\mathcal{Q}\otimes H}\) then for all \(\eta \in H\), the function \(x \mapsto \eta (x)[f(x, \cdot )]\) is in \(\mathcal{E}_{\mathcal{Q}}\).

If \(\mathcal{P} \subseteq \mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) then \(\mathcal{E}_{\mathcal{Q}} \subseteq \mathcal{E}_{\mathcal{P}}\).

Let \(\delta \in [0,1/2]\) and let \(B_{\le \delta } = \{ \operatorname{Ber}(p) \mid p \le \delta \} \). Then

Let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be a Markov kernel and \(H \subseteq (\mathcal{X} \rightsquigarrow \mathcal{Y})\) a set of Markov kernels. If \(\mathcal{X}\) is countable with measurable singletons then there exists a conditional numeraire for \((\kappa , H)\).

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and \(U : \mathbb {R}_{+,\infty } \to \overline{\mathbb {R}}\) a concave function which is bounded from above. Then there exists \(f^* \in \mathcal{E}_{\mathcal{Q}}\) such that

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and \(U : \mathbb {R}_{+,\infty } \to \overline{\mathbb {R}}\) a concave function which is bounded from above. Then there exists \(f^* \in \mathcal{E}_{\mathcal{Q}}\) such that

and such that all e-variables are \(P\)-a.s. finite on \(\{ f^* {\lt} \infty \} \).

Let \((X_n)_{n \in \mathbb {N}}\) be a sequence of measurable functions from \(\mathcal{X}\) to \(\mathbb {R}_{+,\infty }\) and let \(Q \in \mathcal{P}(\mathcal{X})\). There exists a sequence of measurable functions \(\tilde{X}_n \in \mathrm{conv}(X_n, X_{n+1}, \ldots )\) and \(X : \mathcal{X} \to \mathbb {R}_{+, \infty }\) such that \(\tilde{X}_n \to X\) \(Q\)-almost surely.

Let \(f^*\) be a numeraire for \((P, \mathcal{Q})\) and let \(f \in \mathcal{E}_{\mathcal{Q}}\). Then \(P\left[\log \frac{f}{f^*}\right] \le 0\).

Let \(f^*\) be a numeraire for \((P, \mathcal{Q})\). Then for all \(f \in \mathcal{E}_{\mathcal{Q}}\),

Let \(f^*\) be a numeraire for \((P, \mathcal{Q})\). Then \(P[\log f^*] \ge 0\).

Let \(U : \mathbb {R}_{+,\infty } \to \overline{\mathbb {R}}\) be a utility function and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) a set of probability measures. Let \(P \in \mathcal{P}(\mathcal{X})\). Then

Let \(\mathcal{Q}' \subseteq \mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). Then

Let \(\delta \in [0,1/2]\) and let \(B_{\le \delta } = \{ \operatorname{Ber}(p) \mid p \le \delta \} \). Then

Let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be a Markov kernel, \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). Then

Let \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). If \(f^*\) is a numeraire for \((P, \mathcal{Q})\) then

Suppose that there exists an event \(A\) such that \(P(A) {\gt} 0\) and \(Q(A) = 0\) for all \(Q \in \mathcal{Q}\). That is, we don’t have \(P \ll \mathcal{Q}\). Then

Let \(\phi : \mathcal{X} \to \mathcal{Y}\) be a measurable embedding, \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). Then

Let \(f : \mathcal{X} \to \mathcal{Y}\) be a measurable function, \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). Then

Let \(R \in \mathcal{P}(\mathcal{X})\), \(\mathcal{P} \subseteq \mathcal{P}(\mathcal{X})\) and let \(\phi : \mathcal{X} \to \mathcal{X}\) be a measurable map such that \(\phi \circ \phi = \operatorname{id}\). Then

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and let \(f\) be an E-variable for \(\mathcal{Q}\). Then the set \(\{ f = \infty \} \) is \(\mathcal{Q}\)-null.

\(E^*_{P \otimes \kappa , \mathcal{Q} \otimes \kappa } =_{(P\otimes \kappa ) a.e.} (x, y) \mapsto E^*_{P, \mathcal{Q}}(x)\) for any Markov kernel \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\).

Let \(f\) be a \(P\)-almost surely positive E-variable for a set of probability measures \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). Then \(f\) is a numeraire for \((P, \mathcal{Q})\) if and only if \(P[\log \frac{g}{f}] \le 0\) for all \(g \in \mathcal{E}_{\mathcal{Q}}\).

Let \(\mathcal{Q}_1 \subseteq \mathcal{Q}_2\). If \(f^*\) is a numeraire for \((P, \mathcal{Q}_1)\) and \(f^* \in \mathcal{E}_{\mathcal{Q}_2}\) then \(f^*\) is also a numeraire for \((P, \mathcal{Q}_2)\).

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be a Markov kernel. Then for all \(\xi \in \mathcal{E}_{\kappa \circ \mathcal{Q}}^R\), \(\xi \circ \kappa \in \mathcal{E}^R_{\mathcal{Q}}\).

A Markov kernel \(\eta : \mathcal{X} \rightsquigarrow \mathbb {R}_{+,\infty }\) is a randomized E-variable for a set of probability measures \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) if and only if the measurable function \(f : \mathcal{X} \to \mathbb {R}_{+,\infty }\) defined by \(f(x) = \eta (x)[\operatorname{id}]\) is an E-variable for \(\mathcal{Q}\).

For \(P \in \mathcal{M}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{M}(\mathcal{X})\), there is a unique decomposition \(P = P_{\mathrm{ac}} + P_{\perp }\) with \(P_{\mathrm{ac}} \ll \mathcal{Q}\) and \(P_{\perp } \perp \mathcal{Q}\) (meaning that there exists a \(\mathcal{Q}\)-null set \(A\) with \(P_{\perp }(A^c) = 0\)).

The singular set of \(P\) with respect to \(\mathcal{Q}\) is unique up to \(P\) and \(\mathcal{Q}\)-null sets.

Let \(P, Q \in \mathcal{P}(\mathcal{X})\) be two probability measures with \(P \ll Q\). Then

Let \(P, Q \in \mathcal{P}(\mathcal{X})\). Then for all Markov kernels \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\),

Let \(\mathcal{P},\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be two sets of probability measures. Suppose that there exists a measurable function \(f_0 : \mathcal{X} \to \mathbb {R}_{+,\infty }\) which is less than 1 and \(\delta \in [0,1/2]\) such that

Then

Let \(\mathcal{P},\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be two sets of probability measures. Suppose that there exists a measurable function \(f_0 : \mathcal{X} \to \mathbb {R}_{+,\infty }\) which is less than 1 and \(\delta \in [0,1/2]\) such that

Then

Let \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). If \(P \ll \mathcal{Q}\) then

Let \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\) be a Markov kernel and \(H \subseteq (\mathcal{X} \rightsquigarrow \mathcal{Y})\) a set of Markov kernels. If either \(\mathcal{X}\) is countable or \(\mathcal{Y}\) is countably generated, then there exists a conditional numeraire for \((\kappa , H)\).

Let \(P \in \mathcal{P}(\mathcal{X})\) and \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\). A numeraire for \((P, \mathcal{Q})\) exists and is \(P\)-a.e. unique.

Let \(\mathcal{Q} \subseteq \mathcal{P}(\mathcal{X})\) be a set of probability measures and \(P \in \mathcal{P}(\mathcal{X})\). Let \(U : \mathbb {R}_{+,\infty } \to \overline{\mathbb {R}}\) be a utility function. Then for all Markov kernels \(\kappa : \mathcal{X} \rightsquigarrow \mathcal{Y}\),

For any \(f \in \mathcal{E}_{\mathcal{Q} \otimes H}\), we have

Note that this proves the numeraire property for the candidate \(E^*_{P, \mathcal{Q}}E^*_{P \otimes \kappa , P \otimes H}\), but we don’t know if it is an e-variable, so can’t conclude that it’s the numeraire. As a consequence,

If \(E^*_{P, \mathcal{Q}}E^*_{P \otimes \kappa , \{ P\} \otimes H}\) was an e-variable, then we would have equality.

If a conditional numeraire exists for \((\kappa , H)\), then \(E^*_{P \otimes \kappa , \mathcal{Q} \otimes H} =_{a.e.} E^*_{P, \mathcal{Q}}E^*_{\kappa , H}\).

Let \(P_1 \in \mathcal{P}(\mathcal{X}), P_2 \in \mathcal{P}(\mathcal{Y})\) and \(\mathcal{Q}_1 \subseteq \mathcal{P}(\mathcal{X}), \mathcal{Q}_2 \subseteq \mathcal{P}(\mathcal{Y})\). Then the numeraire for the product \((P_1 \times P_2, \mathcal{Q}_1 \times \mathcal{Q}_2)\) is given by the product of the numeraires for \((P_1, \mathcal{Q}_1)\) and \((P_2, \mathcal{Q}_2)\):

Its logarithmic utility is given by