4. Borel–Cantelli Lemma(s).Â Let be a probability space. Then

If , then .

If are independent and , then for -a.e. , .

The dynamical version often involves the orbits of points, instead of the static points. In particular, let be a measure-preserving map on . Then

– is said to be a Borel–Cantelli sequence with respect to if ;

– is said to be a strong Borel–Cantelli sequence if for -a.e. .

3. Let be a Hamiltonian function, be the generating function in the sense that . Then the *Hamiltonâ€“Jacobi equation* is a first-order, non-linear partial differential equation

.

Note that the total derivative . Therefore, is the classical action function (up to an undetermined constant).

2. Let be a family of geodesic on a Riemannian manifold . Then defines a vector field along , which is called a *Jacobi field*. describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic .

Alternatively, A vector field along a geodesic is said to be a *Jacobi field*, if it satisfies the *Jacobi equation*:

where denotes the covariant derivative with respect to the Levi-Civita connection, and the Riemann curvature tensor on .