{
  "id": "enwiki-001272365019",
  "slug": "simple-function",
  "title": "Simple function",
  "qid": "",
  "lead": "In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently \"nice\" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable, as used in practice. A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the \"simple\" of \"simple function\" has a technical meaning somewhat at odds with common language.) All step functions are simple. Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function and also it is straightforward to approximate more general functions by sequences of simple functions.",
  "body": "In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently \"nice\" that using them makes mathematical reasoning, theory, and proof easier. For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable, as used in practice. A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1, 2, 3, 4, 5, 6, 7, 8}. A more advanced example is the Dirichlet function over the real line, which takes the value 1 if x is rational and 0 otherwise. (Thus the \"simple\" of \"simple function\" has a technical meaning somewhat at odds with common language.) All step functions are simple. Simple functions are used as a first stage in the development of theories of integration, such as the Lebesgue integral, because it is easy to define integration for a simple function and also it is straightforward to approximate more general functions by sequences of simple functions. ==Definition== Formally, a simple function is a finite linear combination of indicator functions of measurable sets. More precisely, let (X, Σ) be a measurable space. Let A 1 , ..., A n ∈ Σ be a sequence of disjoint measurable sets, and let a 1 , ..., a n be a sequence of real or complex numbers. A simple function is a function f\\colon X \\to \\mathbb{C} of the form f(x)=\\sum_{k=1}^n a_k {\\mathbf 1}_{A_k}(x), where {\\mathbf 1}_A is the indicator function of the set A. ==Properties of simple functions== The sum, difference, and product of two simple functions are again simple functions, and multiplication by constant keeps a simple function simple; hence it follows that the collection of all simple functions on a given measurable space forms a commutative algebra over \\mathbb{C} . ==Integration of simple functions== If a measure \\mu is defined on the space (X, \\Sigma) , the integral of a simple function f\\colon X \\to \\mathbb R with respect to \\mu is defined to be \\int_X f d \\mu = \\sum_{k=1}^na_k\\mu(A_k), if all summands are finite. ==Relation to Lebesgue integration== The above integral of simple functions can be extended to a more general class of functions, which is how the Lebesgue integral is defined. This extension is based on the following fact. Theorem. Any non-negative measurable function f\\colon X \\to\\mathbb{R}^{+} is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. It is implied in the statement that the sigma-algebra in the co-domain \\mathbb{R}^{+} is the restriction of the Borel σ-algebra \\mathfrak{B}(\\mathbb{R}) to \\mathbb{R}^{+} . The proof proceeds as follows. Let f be a non-negative measurable function defined over the measure space (X, \\Sigma,\\mu) . For each n\\in\\mathbb N , subdivide the co-domain of f into 2^{2n}+1 intervals, 2^{2n} of which have length 2^{-n} . That is, for each n , define I_{n,k}=\\left[\\frac{k-1}{2^n},\\frac{k}{2^n}\\right) for k=1,2,\\ldots,2^{2n} , and I_{n,2^{2n}+1}=[2^n,\\infty) , which are disjoint and cover the non-negative real line ( \\mathbb{R}^{+} \\subseteq \\cup_{k}I_{n,k}, \\forall n \\in \\mathbb{N} ). Now define the sets A_{n,k}=f^{-1}(I_{n,k}) \\, for k=1,2,\\ldots,2^{2n}+1, which are measurable ( A_{n,k}\\in \\Sigma ) because f is assumed to be measurable. Then the increasing sequence of simple functions f_n=\\sum_{k=1}^{2^{2n}+1}\\frac{k-1}{2^n}{\\mathbf 1}_{A_{n,k}} converges pointwise to f as n\\to\\infty . Note that, when f is bounded, the convergence is uniform. ==See also== Bochner measurable function",
  "categories": [
    "22946561",
    "23118434",
    "23603337",
    "24745114",
    "29176493",
    "29176591",
    "29176689"
  ],
  "license": "CC BY-SA 4.0",
  "source_uri": "https://en.wikipedia.org/w/index.php?oldid=1272365019",
  "content_hash": "",
  "created_at": null,
  "updated_at": 1738046169.0,
  "claims": [],
  "contradictions": [],
  "confidence": 0.6,
  "csl": {
    "id": "enwiki-001272365019",
    "type": "webpage",
    "title": "Simple function",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/simple-function",
    "issued": {
      "date-parts": [
        [
          2026,
          11,
          12
        ]
      ]
    },
    "accessed": {
      "date-parts": [
        [
          2026,
          11,
          12
        ]
      ]
    },
    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001272365019",
    "isBasedOn": "https://en.wikipedia.org/w/index.php?oldid=1272365019",
    "note": "Immutable snapshot enwiki-001272365019"
  }
}