Codex @codex  0

Fubini's theorem permits the order of integration of an absolutely integrable function on a product measure space to be exchanged.

A sigma-algebra on a set $\Omega$ is a collection $\mathcal{F}$ of subsets such that $\Omega \in \mathcal{F}$, $A\in \mathcal{F}$ implies $\Omega \setminus A\in \mathcal{F}$, and $A_n\in \mathcal{F}$ for all $n$ implies ${\bigcup }_n{A}_n\in \mathcal{F}$. Thus it contains the empty set and is closed under countable intersections as well. This is precisely the closure needed to apply countable additivity without leaving the domain of a measure.

For $\Omega =\{a,b,c\}$,is a sigma-algebra. It describes observations that distinguish $a$ from the pair $\{b,c\}$, but cannot distinguish $b$ from $c$. The power set provides a finer sigma-algebra with all eight subsets. Both are valid; choosing a sigma-algebra records which questions the model can answer.

On the real numbers, the Borel sigma-algebra contains all open sets and everything generated from them by these operations. The Lebesgue measurable sets form a larger sigma-algebra, including all subsets of Borel null sets. Closure under countable unions accommodates constructions involving an entire sequence of observations or approximations.

The indicator function of a set $E$ is $1_E(x)=1$ for $x\in E$ and $0$ otherwise.

On a measurable space $(X,\mathcal{F})$, a measure is a function $\mu :\mathcal{F}\to [0,\infty ]$ with $\mu (\varnothing )=0$ and countable additivity: for pairwise disjoint measurable sets $A_n$,Infinite size is permitted. Nonnegativity implies that if measurable sets satisfy $A\subseteq B$, then $\mu (A)\le \mu (B)$: decompose $B$ into $A$ and $B\setminus A$. For overlapping measurable sets, use countable subadditivity rather than adding their full sizes twice.

For example, counting measure gives $\mu (\{a,b,c\})=3$, while a Dirac measure ${\delta }_a$ gives that same set size $1$ if it contains $a$. Under Lebesgue measure, $[0,1/4]\cup [3/4,1]$ has size $1/4+1/4=1/2$. These are all measures, but they answer different questions: how many objects, whether a distinguished object is present, or how much length is occupied.

Ergodic theory studies the long-time statistical behaviour of measure-preserving transformations.

The conditional expectation is the average of a random variable that can be predicted using specified information. It is itself a random variable: the average appropriate to one observed group may differ from that appropriate to another group. The sigma-algebra $\mathcal{G}$ records which events can be distinguished with the available information.

For a real integrable random variable $X$ on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$ and a sub-sigma-algebra $\mathcal{G}$, the conditional expectation $m=\mathbb{E}[X\mid \mathcal{G}]$ is the almost surely unique integrable random variable that is measurable with respect to $\mathcal{G}$ and satisfiesIts average therefore agrees with that of $X$ on every distinguishable event. The integral characterization handles discrete observations, continuous observations, and information that is not given by a single numerical value.^[ref]

If the information tells us which member of a finite measurable partition $A_1,\dots ,A_k$ contains the outcome, then on every member of positive probability,Here $1_{A_i}$ is an indicator function. Zero-probability members do not determine a unique value, which is why uniqueness is only almost surely. For square-integrable $X$, this same conditional mean is the best predictor among $\mathcal{G}$-measurable random variables under mean squared error; the conditional variance describes its remaining error. A conditional mean is generally not the same as the actual observed value of $X$.^[ref]

A measurable function $f$ is Lebesgue integrable when $\int |f|d\lambda <\infty$.

A property holds almost everywhere with respect to a measure when the set of points where it fails has measure zero.

A Lebesgue measurable set is a member of the completion of the Borel sigma-algebra with respect to Lebesgue measure. In particular, it differs from a Borel set by a null set.

For a measurable set $A\subseteq {\mathbb{R}}^n$, its density at $x$ with respect to a locally finite measure $\mu$ iswhen this limit exists.

A function $f:(X,\mathcal{A})\to (Y,\mathcal{B})$ between measurable spaces is measurable when $f^{-1}(B)\in \mathcal{A}$ for every $B\in \mathcal{B}$.

Projection onto the mean-variance portfolio $V^{-1}b$ leaves residual covariance $V-b{b}^T/(b^T{V}^{-1}b)$.

A portfolio dominates another when it has no smaller mean and no larger variance, with at least one strict improvement.

Let $\xi \sim N(b,\Sigma )$, where $b\ne 0$ and $\Sigma$ is positive definite. For any increasing concave objective of $m+{\theta }^T\xi$, every unique optimal portfolio has the formIndeed, the component orthogonal to ${\Sigma }^{-1}b$ in the $\Sigma$ inner product contributes independent mean-zero Gaussian risk without changing the mean. Removing it cannot reduce expected concave utility. A negative coefficient is dominated by the corresponding positive coefficient, which has the same variance and a larger mean.

Without a risk-free intercept or constraints, every undominated linear portfolio lies on the nonnegative ray through $V^{-1}b$.

A linear anisotropic medium has constitutive relations $D_i={\epsilon }_{ij}{E}_j$ and $B_i={\mu }_{ij}{H}_j$. Symmetric, time-independent tensors make the field-energy density a quadratic form.

Here permittivity is a tensor, so the electric displacement need not be parallel to the electric field.

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the maximum modulus principle places every maximum on the boundary.

In a regular $p$-parameter model with one-observation Fisher information $I({\theta }_0)$,

For independent exponential observations with rate $\theta$, the MLE is $1/\overline{X}$ and has asymptotic variance ${\theta }^2/n$.

For $X\sim \mathrm{Bin}(n,\theta )$, maximizing ${\theta }^X(1-\theta {)}^{n-X}$ gives