Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
Σ-finite measureIn mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.
Greenstick fractureA greenstick fracture is a fracture in a young, soft bone in which the bone bends and breaks. Greenstick fractures occur most often during infancy and childhood when bones are soft. The name is by analogy with green (i.e., fresh) wood which similarly breaks on the outside when bent. Some clinical features of a greenstick fracture are similar to those of a standard long bone fracture - greenstick fractures normally cause pain at the injured area.
Human skeletonThe human skeleton is the internal framework of the human body. It is composed of around 270 bones at birth – this total decreases to around 206 bones by adulthood after some bones get fused together. The bone mass in the skeleton makes up about 14% of the total body weight (ca. 10–11 kg for an average person) and reaches maximum mass between the ages of 25 and 30. The human skeleton can be divided into the axial skeleton and the appendicular skeleton. The axial skeleton is formed by the vertebral column, the rib cage, the skull and other associated bones.
Complete measureIn mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (X, Σ, μ) is complete if and only if The need to consider questions of completeness can be illustrated by considering the problem of product spaces. Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by We now wish to construct some two-dimensional Lebesgue measure on the plane as a product measure.
Bone marrowBone marrow is a semi-solid tissue found within the spongy (also known as cancellous) portions of bones. In birds and mammals, bone marrow is the primary site of new blood cell production (or haematopoiesis). It is composed of hematopoietic cells, marrow adipose tissue, and supportive stromal cells. In adult humans, bone marrow is primarily located in the ribs, vertebrae, sternum, and bones of the pelvis. Bone marrow comprises approximately 5% of total body mass in healthy adult humans, such that a man weighing 73 kg (161 lbs) will have around 3.
OsteopeniaOsteopenia, known as "low bone mass" or "low bone density", is a condition in which bone mineral density is low. Because their bones are weaker, people with osteopenia may have a higher risk of fractures, and some people may go on to develop osteoporosis. In 2010, 43 million older adults in the US had osteopenia. Unlike osteoporosis, osteopenia does not usually cause symptoms, and losing bone density in itself does not cause pain.
Bone marrow adipose tissueBone marrow adipose tissue (BMAT), sometimes referred to as marrow adipose tissue (MAT), is a type of fat deposit in bone marrow. It increases in states of low bone density -osteoporosis, anorexia nervosa/ caloric restriction, skeletal unweighting such as that which occurs in space travel, and anti-diabetes therapies. BMAT decreases in anaemia, leukaemia, and hypertensive heart failure; in response to hormones such as oestrogen, leptin, and growth hormone; with exercise-induced weight loss or bariatric surgery; in response to chronic cold exposure; and in response to pharmacological agents such as bisphosphonates, teriparatide, and metformin.
Bone mineralBone mineral (also called inorganic bone phase, bone salt, or bone apatite) is the inorganic component of bone tissue. It gives bones their compressive strength. Bone mineral is formed predominantly from carbonated hydroxyapatite with lower crystallinity. Bone mineral is formed from globular and plate structures distributed among the collagen fibrils of bone and forming yet a larger structure. The bone salt and collagen fibers together constitute the extracellular matrix of bone tissue.
Borel measureIn mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Let be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of ; this is known as the σ-algebra of Borel sets. A Borel measure is any measure defined on the σ-algebra of Borel sets.
