Locally integrable function

Key takeaways

Locally integrable function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Locally integrable function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Locally integrable function from memory before moving on to harder problems.

Reference excerpt

In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to p-integrable function spaces ( L p {\textstyle L^{p}} spaces), but its members are not required to satisfy any growth restriction on their behaviour at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.

Definition

Standard definition Definition 1. Let Ω {\textstyle \Omega } be an open set in the Euclidean space R n {\textstyle \mathbb {R} ^{n}} and f : Ω → C {\textstyle f:\Omega \to {\mathbb {C}}} be a Lebesgue measurable function. If f {\textstyle f} on Ω {\textstyle \Omega } is such that

∫ K | f | d x < + ∞ , {\displaystyle \int _{K}|f|\,\mathrm {d} x<+\infty ,}

i.e. its Lebesgue integral is finite on all compact subsets K {\textstyle K} of Ω {\textstyle \Omega } , then f {\textstyle f} is called locally integrable. The set of all such functions is denoted by L 1 , loc ( Ω ) {\textstyle L_{1,{\text{loc}}}(\Omega )} :

L 1 , l o c ( Ω ) = { f : Ω → C measurable : f | K ∈ L 1 ( K ) ∀ K ⊂ Ω , K compact } , {\displaystyle L_{1,\mathrm {loc} }(\Omega )={\bigl \{}f\colon \Omega \to \mathbb {C} {\text{ measurable}}:f|_{K}\in L_{1}(K)\ \forall \,K\subset \Omega ,\,K{\text{ compact}}{\bigr \}},}

where f | K {\textstyle \left.f\right|_{K}} denotes the restriction of f {\textstyle f} to the set K {\textstyle K} .

An alternative definition Definition 2. Let Ω {\textstyle \Omega } be an open set in the Euclidean space R n {\textstyle \mathbb {R} ^{n}} . Then a function f : Ω → C {\textstyle f:\Omega \to \mathbb {C} } such that

∫ Ω | f φ | d x < + ∞ , {\displaystyle \int _{\Omega }|f\varphi |\,\mathrm {d} x<+\infty ,}

for each test function φ ∈ C c ∞ ( Ω ) {\textstyle \varphi \in C_{c}^{\infty }(\Omega )} is called locally integrable, and the set of such functions is denoted by L 1 , loc ( Ω ) {\textstyle L_{1,{\text{loc}}}(\Omega )} . Here, C c ∞ ( Ω ) {\textstyle C_{c}^{\infty }(\Omega )} denotes the set of all infinitely differentiable functions φ : Ω → R {\textstyle \varphi \colon \Omega \to {\mathbb {R}}} with compact support contained in Ω {\textstyle \Omega } . This definition has its roots in the approach to measure and integration theory based on the concept of a continuous linear functional on a topological vector space, developed by the Nicolas Bourbaki school. It is also the one adopted by Strichartz (2003) and by Maz'ya & Shaposhnikova (2009, p. 34). This "distribution theoretic" definition is equivalent to the standard one, as the following lemma proves: Lemma 1. A given function f : Ω → C {\textstyle f:\Omega \to \mathbb {C} } is locally integrable according to Definition 1 if and only if it is locally integrable according to Definition 2, i.e.,

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Locally integrable function

Start with the simplest possible case. Write down what Locally integrable function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Locally integrable function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Locally integrable function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Locally integrable function

In research: Locally integrable function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Locally integrable function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Locally integrable function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integral calculus, Lp spaces, Measure theory, so understanding it makes those chapters shorter.
In everyday life: Look for Locally integrable function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Locally integrable function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Locally integrable function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Locally integrable function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Locally integrable function in simple terms?

In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar t…

Why does Locally integrable function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Locally integrable function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Locally integrable function.