The Moreau-Yosida regularization is a technique used to approximate lower semicontinuous functions by Lipschitz functions. An important application of this result is to prove Portmanteau's Theorem, which states that integration against a lower semicontinuous and bounded below function is lower semicontinuous with respect to the narrow convergence in the space of probability measures.
Let be a metric space, and let denotes the collection of probability measures on . is said to be a Polish space if it is complete and separable.
A function is said to be proper  if it is not identically equal to , that is, if there exists such that . The domain of is the set
For a given function and , its Moreau-Yosida regularization  is given by
The distance term may often be raised to a positive exponent , in particular . For example, when is a Hilbert space  , is taken to be
This particular variant in a Hilbert space setting is explored in more detail below.
The dependence on the parameter may also be written instead as
- If , then by definition is constant and .
- If is not proper, then for all .
Take . If is finite-valued and differentiable, we can write down an expression for . For a fixed , the map is continuous everywhere and differentiable everywhere except for when , where the derivative does not exist due to the absolute value. Thus we can apply standard optimization techniques from Calculus to solve for : find the critical points of and take the infimum of evaluated at the critical points. One of these values will always be the original function evaluated at , since this corresponds to the critical point for .
- Let . Then
Approximating Lower Semicontinuous Functions by Lipschitz Functions
Proposition.  Let be a Polish space and let .
- If is proper and bounded below, so is . Furthermore, is Lipschitz continuous for all .
- If, in addition, is lower semicontinuous, then for all .
- In this case, is continuous and bounded and for all .
- Since is proper, there exists such that . Then for any
Thus is proper and bounded below. Next, for a fixed , let . Then as
the family is uniformly Lipschitz and hence equicontinuous. Thus is Lipschitz continuous.
- Suppose that is also lower semicontinuous. Note that for all , . Thus it suffices to show that . This inequality is automatically satisfied when the left hand side is infinite, so without loss of generality assume that . By definition of infimum, for each there exists such that
is bounded below by assumption, while the only way to be finite in the limit is for to vanish in the limit. Thus converges to in , and by lower semicontinuity of ,
- By definition, . Since for all , for all .
Theorem (Portmanteau).   Let be a Polish space, and let be lower semicontinuous and bounded below. Then the functional is lower semicontinuous with respect to narrow convergence in , that is
Proof. By the Moreau-Yosida approximation, for all ,
Taking , Fatou's Lemma ensures that
Etymology of Portmanteau Theorem
The curious epithet attached to the above theorem is due to Billingsley , with a citation to a Jean-Pierre Portmanteau's Espoir pour l'ensemble vide? published in Annales de l'Université de Felletin in 1915. This is believed to be a fictional citation made as a play on words .
- The publication date is far too early; Kolmogorov's probability axioms were published in 1933. 
- Felletin is a small town in central France with no university, and there is no record of a Jean-Pierre Portmanteau aside from this citation.
- "Espoir pour l'ensemble vide" translates to "hope for the empty set" (translation was by Google, please confirm or amend if you speak French!)
The Moreau-Yosida regularization is a specific case of a type of convolution, and many of the above results follow from this generalization. This material is adapted from Bauschke-Combettes Chapter 12 , where the setting is over a Hilbert space instead of a more general Polish space.
Let be a Hilbert space, and let . The infimal convolution or epi-sum of and is
is said to be exact at a point if this infimum is attained. is said to be exact if it is exact at every point of its domain, and in this case it is denoted by .
Remark. Bauschke-Combettes uses a box with a dot in the middle for to be exact. Due to technical difficulties, we will use instead.
For an example, let be nonempty. Then is exact, and .
Proposition. Let be proper, , and for , let be given by
Then the following hold for all and :
- for , ,
- as , and
- is bounded above on every ball in .
Remark. The convention given above differs slightly from Bauschke-Combettes to fit the convention in this article. The Moreau-Yosida regularization is the special case where , and is called the Pasch-Hausdorff Envelope in Bauschke-Combettes.
Proposition. Let be lower semicontinuous and convex, let , and let . Then the infimal convolution is convex, proper, continuous, and exact. Moreover, for every , the infimum
is uniquely attained.
- ↑ 1.0 1.1 1.2 1.3 Craig, Katy C. Lower Semicontinuity in the Narrow Topology. Math 260J. Univ. of Ca. at Santa Barbara. Winter 2022.
- ↑ 2.0 2.1 Bauschke, Heinz H. and Patrick L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd Ed. Ch. 12. Springer, 2017.
- ↑ Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Ch. 3.1. Birkhäuser, 2005.
- ↑ 4.0 4.1 Santambrogio, Filippo. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling Ch. 1.1. Birkhäuser, 2015.
- ↑ Billingsley, Patrick. Convergence of Probability Measures, 2nd Ed. John Wiley & Sons, Inc. 1999.
- ↑ Pagès, Gilles. Numerical Probability: An Introduction with Applications to Finance. Ch. 4.1. Springer, 2018.
- ↑ Kolmogorov, Andrey (1950) . Foundations of the theory of probability. New York, USA: Chelsea Publishing Company.