A measurable loss can have an infinite normalizer

Given: The von Neumann natural numbers $\mathbb{N}$, their power-set sigma-algebra, weights $a_k=(1\mathrm{/}2{)}^{k+1}$, and Dirac measures ${\delta }_k$.

Generalized Bayes loss update and temperature: at fixed $w>0$, $Z=\int e^{-w\mathrm{\ell }}d\pi$ must be finite and positive for the update to be defined.

A countable discrete measurable space is standard Borel: every nonempty at-most-countable set with its power-set sigma-algebra is standard Borel.

Finite, countably infinite, countable, uncountable: a set equinumerous with $\mathbb{N}$ is at-most-countable.

The natural numbers $\mathbb{N}$ (von Neumann): $0\in \mathbb{N}$ and $\mathbb{N}$ is closed under successor.

Nonnegative scalar multiples and countable weighted sums of measures are measures: a countable nonnegative weighted sum of measures on one measurable space is a measure.

The Dirac set function at a point: ${\delta }_k(A)=1$ when $k\in A$ and is zero otherwise.

A Dirac set function is a probability measure: each ${\delta }_k$ is a probability measure.

Probability measures and probability spaces: a probability measure has total mass one.

The principle of mathematical induction: a property holding at zero and preserved by successor holds for all natural numbers.

Integer powers $a^m$: $b^0=1$ and $b^{k+1}=b^{k}b$ for natural $k$.

For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges: for $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{j=0}^{\mathrm{\infty }}{r}^j=1\mathrm{/}(1-r)$.

The natural logarithm as the inverse of the exponential function: $\log$ is inverse to $\exp$ on $(0,\mathrm{\infty })$.

The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$: $\exp (t)>0$ and $\exp (-t)=1\mathrm{/}\exp (t)$.

Nonnegative simple measurable functions: a finite nonnegative linear combination of indicators of measurable sets is a nonnegative simple measurable function.

The nonnegative Lebesgue integral: the nonnegative integral is the supremum of the integrals of nonnegative simple minorants, with values in $[0,+\mathrm{\infty }]$.

The integral of a nonnegative simple function: the integral of a simple function represented on disjoint measurable sets is the sum of its coefficient times the measure of each set.

The nonnegative integral agrees with the simple integral on simple functions: the nonnegative integral of a simple function equals its simple integral.

Monotonicity and nonnegative homogeneity of the nonnegative integral: if $0\le f\le g$, then $\int fd\pi \le \int gd\pi$.

The Cauchy-sequence reals have the least-upper-bound property: $\mathbb{R}$ is a complete ordered field.

Every complete ordered field is Archimedean: in a complete ordered field, for every $x$ there is a natural $n\ge 1$ with $x<n$.

Absolute value in an ordered field: if $b\ge 0$, then $\mathrm{\mid }b\mathrm{\mid }=b$.

Ordered field: products of positive reals are positive.

The multiplicative identity is positive: $1>0$.

Laws of finite sums and finite products: finite sums scale by a real constant, ${\sum }_{k<n}c{u}_k=c{\sum }_{k<n}{u}_k$.

Series, partial sums, convergence and the sum, divergence, and the tail series: a convergent series sums to the limit of its finite partial sums.

Algebra of limits: sums, scalar multiples, products and quotients: scaling a convergent sequence by a fixed real constant scales its limit by that constant.

A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum: for nonnegative terms, the usual convergent series sum is the supremum of its finite partial sums.

Series in the nonnegative extended real line: the nonnegative extended sum is the supremum of its finite partial sums.

Nonnegative scalar multiples and countable weighted sums of measures: the countable weighted sum is evaluated pointwise by the nonnegative extended sum of its weighted measures.