Measure Theory Minimum

Quantum measurement probabilities are measures on sets of outcomes, and an expectation is an integral with respect to such a measure. This distinction becomes indispensable when a spectrum has continuous or singular parts: a probability density need not exist, and a normalized state need not have a finite moment of an unbounded observable. The minimum needed here is the ability to specify measurable events, identify the measure, and test the relevant integrability condition.

Helpful background. L2 Spaces explains equality almost everywhere. Projection-Valued Measures supplies the quantum observable whose scalar probabilities we analyze below.

Outcome events and scalar spectral measures

A sigma-algebra is a collection of subsets closed under complements and countable unions. A positive measure assigns nonnegative sizes to these sets, with countable additivity on disjoint unions. For a real-valued sharp observable, the outcome events are usually the Borel sets of $\mathbb R$, generated by open intervals.

If $A$ is self-adjoint with spectral measure $E_A$ and $\|\psi\|=1$, the rule

$$\mu_\psi^A(\Delta)=\langle\psi,E_A(\Delta)\psi\rangle$$

is a probability measure. Positivity follows because spectral projections are positive; normalization follows from $E_A(\mathbb R)=I$. Strong countable additivity of the projections implies scalar countable additivity after taking this matrix element. No eigenbasis or density is required for the definition.

A point mass $\mu_\psi^A(\{a\})$ is the probability of the exact value $a$. A continuous distribution may assign zero probability to every singleton while assigning probability one to their uncountable union. Countable additivity never permits replacing that union by an uncountable sum of zeros.

Densities and almost-everywhere statements

A probability density $p$ relative to Lebesgue measure exists when the probability measure is absolutely continuous with respect to it. The Radon–Nikodym theorem then gives

$$\mu(\Delta)=\int_\Delta p(x)\,dx.$$

The choice of reference measure matters. A discrete energy distribution is naturally a weighted counting measure, not an ordinary Lebesgue density. Singular continuous measures provide a third possibility; the spectral theorem accommodates them even when elementary models do not require them.

Wavefunctions in $L^2(\mathbb R,dx)$ are equivalence classes modulo sets of Lebesgue measure zero. Multiplication probabilities $\int_\Delta|\psi(x)|^2dx$ are well defined on those classes. A single value $\psi(x_0)$ is not. This is why assigning point values is an extra regularity question, not part of square integrability.

Moments and operator domains are different integrability tests

For a real random outcome $\lambda$, a finite expectation requires

$$\int |\lambda|\,d\mu_\psi^A(\lambda)<\infty.$$

The operator vector $A\psi$ exists in the Hilbert space under the stronger condition

$$\psi\in D(A) \quad\Longleftrightarrow\quad \int \lambda^2\,d\mu_\psi^A(\lambda)<\infty.$$

The first condition is equivalent to $\psi\in D(|A|^{1/2})$. When it holds, the mean can be defined spectrally even if the expression $\langle\psi,A\psi\rangle$ is not literally an inner product with a Hilbert-space vector $A\psi$. A conditionally convergent symmetric cutoff integral is not a finite probabilistic expectation.

For a concrete example, let $Ne_n=ne_n$ on $\ell^2(\mathbb N)$ and take

$$\psi=C\sum_{n=1}^{\infty}n^{-5/4}e_n, \qquad C^{-2}=\sum_{n=1}^{\infty}n^{-5/2}.$$

The spectral mean is $C^2\sum_n n^{-3/2}<\infty$, whereas the second moment is $C^2\sum_n n^{-1/2}=\infty$. This state has a finite mean of $N$ but does not belong to $D(N)$. The Unbounded Spectral Theorem gives the general domain construction.

Which convergence theorem justifies the calculation?

Monotone convergence applies to increasing nonnegative measurable functions. It permits infinite values and is well suited to constructing positive moments by spectral cutoffs.

Dominated convergence applies when functions converge almost everywhere and their absolute values are bounded by one fixed integrable function. It often justifies bounded spectral functions, since a scalar spectral probability measure has finite mass. For example, $e^{-it\lambda}\to1$ as $t\to0$ and $|e^{-it\lambda}-1|^2\le4$ prove strong continuity of the unitary group through a scalar integral. Differentiating requires an extra integrable bound involving $\lambda^2$.

Tonelli’s theorem permits exchange of integrals for nonnegative functions on sigma-finite product measure spaces, possibly with infinite result. Fubini’s theorem permits exchange for absolutely integrable functions and yields finite iterated integrals almost everywhere. An oscillatory real-time path-integral expression does not become a positive measure integral merely because it resembles this notation.

These are imported measure-theory theorems. Teschl’s integration appendix supplies their construction and hypotheses; the quantum application is the choice of spectral measure and integrable majorant.

Exercise: normalization does not imply finite energy

For the same $N$, choose $\psi=C\sum_{n\ge1}n^{-1}e_n$. Determine whether it is normalized after choosing $C$, whether its energy mean is finite, and whether it belongs to $D(N)$.

Solution

The norm is finite since $\sum n^{-2}<\infty$, so $C$ can normalize the state. The mean is $C^2\sum n^{-1}=\infty$ and the second moment is $C^2\sum1=\infty$. Thus it is a valid state vector with an infinite positive energy expectation in this model, and it lies outside the operator domain. Normalization answers a probability question; moment conditions answer additional questions about an observable.

References

• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — spectral measures and unbounded observables.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s edition — Borel measures, convergence theorems, and spectral integration.