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The present paper originated from an attempt to solve a problem of Maharam (see [5]). She asked whether a sequentially point continuous outer measure defined on a 0-algebra °~ of subsets of some set X can be controlled by a countably additive probability measure (i.e. whether there exists a countably additive probability measure with the same null sets). Our results may be considered as a partial solution of this problem, since we give a "'counterexample" which is ~-subadditive but not necessarily sequentially point continuous.The result is applied to the construction of an abelian topological group with very strange properties.First we fix some terminology. An algebra of sets is a Boolean algebra d of subsets of some set X. A real valued (finite) set function (p defined on d is called weakly finitely (countably) subadditive if q)(uAi) < Z (Ai) for any finite (countable) family A i (i~I) of disjoint sets in ,~¢. If this inequality holds for not necessarily disjoint families of sets we call the set function q~ finitely or countably subadditive. Instead of "finitely subadditive" we shall usually write simply subadditive and instead of "countably subadditive" ¢-subadditive.A weakly subadditive (Q-subadditive) set function q~ on .~¢ is called a submeasure (O-submeasure) if its values are non negative, it is increasing (i.e. for A c B we have always q~ (A)< q~ (B)) and ~o (0)= 0. It is clear that a submeasure is subadditive. All Q-submeasures are automatically countably subadditive.All measures considered in the present paper are finitely additive, real valued and non negative set functions. A submeasure q~ on d is called pathological if it is not identically zero and there does not exist a non trivial measure u on d dominated by q~, Let,~-be a paving of subsets of X such that there exist at least one finite covering of X with W-sets. Let ~ be a non negative real valued setfunction defined on ,~-. Then the set function ~0, defined for all subsets A of X aswhere the infimum is taken over all finite coverings of A with ,~'-sets, is a submeasure defined on the Boolean algebra ~( X ) of all subsets of X.It is easily seen that any submeasure can be obtained in this way. By this procedure any submeasure defined on d can be extended to the algebra of all subsets of X. Such an extension is of course maximal (in the pointwise ordering) between all extensions.

Background COVID‐19 is known to cause an acute respiratory illness, although clinical manifestations outside of the respiratory tract may occur. Early reports have identified SARS‐CoV‐2 as a cause of subacute thyroiditis (SAT). Methods A systematic review was conducted in accordance with the Preferred Reporting Items for Systematic Reviews and Meta‐Analyses (PRISMA) guidelines. MEDLINE, Web of Science and PubMed databases were queried in February 2021 for studies from December 2019 to February 2021. MeSH search terms ‘COVID‐19’, ‘SARS‐CoV‐2’ and ‘coronavirus’ along with search terms ‘thyroiditis’, ‘thyrotoxicosis’ and ‘thyroid’ were used. Descriptive statistics for continuous variables and proportions for categorical variables were calculated. Results Fifteen publications reporting on 17 individual cases of COVID‐19‐induced SAT were identified. Age ranged from 18 to 69 years. The majority (14 of 17; 82%) of cases were female. The delay between onset of respiratory symptoms and diagnosis of SAT ranged from 5 to 49 days (mean, 26.5). Systemic inflammatory response syndrome related to viral infection was uncommonly reported at the time of SAT diagnosis. Thyroid ultrasonography frequently reported an enlarged hypoechoic thyroid with decreased vascularity and heterogenous echotexture. Elevated C‐reactive protein (CRP) was common at the time of SAT diagnosis, with results ranging from 4.5 to 176 mg/L (mean, 41 mg/L). Antithyroid antibodies were frequently negative. SAT‐specific treatment included corticosteroids for 12 of 17 (70.5%) patients. Most returned to normal thyroid status. Conclusion COVID‐19‐associated SAT may be difficult to identify in a timely manner due to potential absence of classic symptoms, as well as cross‐over of common clinical features between COVID‐19 and thyrotoxicosis.

High-energy blunt or penetrating impact leads to great variability in facial injury patterns. Although the mechanism, pattern, and distribution of forces vary, the resultant damage to hard and soft tissues requires dedicated planning and execution of debridement and reconstructive procedures. This article evaluates the initial management of patients sustaining high-energy facial impact injuries resulting in one or more comminuted or displaced facial fractures, with accompanying severe facial lacerations and/or soft tissue defects and avulsion injuries. Seventy-three patients met the criteria for high-energy traumatic injuries at Jackson Memorial/University of Miami Medical Center between 2003 and 2013 and are included in this article. Thirty-nine patients sustained one or more gunshot wounds to the face, and 34 patients were involved in high-speed motor vehicle collisions; all patients met our criteria for high-energy trauma. The treatment protocol for these injuries involves meticulous surgical exploration and assessment, aggressive debridement, early definitive reduction/fixation, and reconstruction as necessary.

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