Measurable Function - Special Measurable Functions

Special Measurable Functions

  • If (X, Σ) and (Y, Τ) are Borel spaces, a measurable function f: (X, Σ) → (Y, Τ) is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin's theorem. If a Borel function happens to be a section of some map, it is called a Borel section.
  • A Lebesgue measurable function is a measurable function, where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers C. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated.
  • Random variables are by definition measurable functions defined on sample spaces.

Read more about this topic:  Measurable Function

Famous quotes containing the words special and/or functions:

    “Our normal waking consciousness, rational consciousness as we call it, is but one special type of consciousness, whilst all about it, parted from it by the filmiest of screens, there lie potential forms of consciousness entirely different.”
    —William James (1842–1910)

    “Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
    —Henry David Thoreau (1817–1862)