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Atom (measure theory)

In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller but positive measure. A measure which has no atoms is called non-atomic or atomless.

Definition

Given a measurable space (X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if

and for any measurable subset B \subset A with

the set B has measure zero.

Examples

  • Consider the set X={1, 2, ..., 9, 10} and let the sigma-algebra \Sigma be the power set of X. Define the measure \mu of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons {i}, for i=1,2, ..., 9, 10 is an atom.
  • Consider the Lebesgue measure on the real line. This measure has no atoms.
  • Non-atomic measures

    A measure which has no atoms is called non-atomic. In other words, a measure is non-atomic if for any measurable set A with  \mu (A) >0 there exists a measurable subset B of A such that

    A non-atomic measure with at least one positive value has an infinite number of distinct values, as starting with a set A with  \mu (A) >0 one can construct a decreasing sequence of measurable sets

    Atomic (song)

    "Atomic" is a hit song by the American new wave band Blondie, written by Debbie Harry and Jimmy Destri and produced by Mike Chapman. It was released as the third single from the band's Platinum-selling 1979 album Eat to the Beat.

    Song information

    Atomic was composed by Jimmy Destri and Debbie Harry, who (in the book "1000 UK #1 Hits" by Jon Kutner and Spencer Leigh) stated "He was trying to do something like "Heart of Glass", and then somehow or another we gave it the spaghetti western treatment. Before that it was just lying there like a lox. The lyrics, well, a lot of the time I would write while the band were just playing the song and trying to figure it out. I would just be scatting along with them and I would just start going, 'Ooooooh, your hair is beautiful.'" The word atomic in the song carries no fixed meaning and functions as a signifier of power and futurism.

    The song was produced as a mixture of new wave, rock and disco which had proven to be so successful in their No.1 hit from earlier in 1979, "Heart of Glass". It is written in E natural minor ("Call Me" is written in E♭ natural minor).

    Sherpa people

    Sherpa (Tibetan: "eastern people", from shar "east" and pa "people") are an ethnic group from the most mountainous region of Nepal, high in the Himalayas.

    Sherpa as a surname appears to be the result of the Nepalese government census takers. Not recognizing that some people only have one name, they wrote the word on census forms in the space for last name. In some cases the clan name was written and in others the ethnicity, i.e. Sherpa. These have then been adopted or forced to be used as last names, last names not being a part of Sherpa culture. A similar effect is seen with the Karen people, who also do not traditionally have more than one name and are a mobile mountain folk.

    Most Sherpa people live in Nepal's eastern regions; however, some live farther west in the Rolwaling valley and in the Helambu region north of Kathmandu. Tengboche is the oldest Sherpa village in Nepal. The Sherpa language belongs to the south branch of the Tibeto-Burman languages; however, this language is separate from Lhasa Tibetan and unintelligible to Lhasa speakers.

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