# simple function

In measure theory, a *simple function ^{}* is a function that is a
finite linear combination

^{}

$$h=\sum _{k=1}^{n}{c}_{k}{\chi}_{{A}_{k}}$$ |

of characteristic functions^{}, where the ${c}_{k}$ are real coefficients and
every ${A}_{k}$ is a measurable set^{} with respect to a fixed measure space^{}.

If the measure space is $\mathbb{R}$ and each ${A}_{k}$ is an interval,
then the function is called a *step function*. Thus, every step
function is a simple function.

Simple functions are used in analysis^{} to interpolate between
characteristic functions and measurable functions^{}. In other words,
characteristic functions are easy to integrate:

$${\int}_{E}{\chi}_{A}\mathit{d}x=|A|,$$ |

while simple functions are not much harder to integrate:

$${\int}_{E}\sum _{k=1}^{n}{c}_{k}{\chi}_{{A}_{k}}dx=\sum _{k=1}^{n}{c}_{k}|{A}_{k}|.$$ |

To integrate a measurable function, one approximates it from below by simple functions. Thus, simple functions can be used to define the Lebesgue integral over a subset of the measure space.

Title | simple function |

Canonical name | SimpleFunction |

Date of creation | 2013-03-22 12:21:16 |

Last modified on | 2013-03-22 12:21:16 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 9 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 03-00 |

Classification | msc 26A09 |

Classification | msc 26-00 |

Classification | msc 28-00 |

Related topic | CharacteristicFunction |

Related topic | Integral2 |

Defines | step function |