Borel measure

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let $\mathfrak{B}(X)$ be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure. Some authors require in addition that μ(C) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure (some authors also require it to be tight). If μ is both inner regular and locally finite, it is called a Radon measure. Note that a locally finite Borel measure automatically satisfies μ(C) < ∞ for every compact set C.

On the real line

The real line $\mathbb{ℝ}$ with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. In this case, $\mathfrak{B}(\mathbb{ℝ})$ is the smallest σ-algebra that contains the open intervals of $\mathbb{ℝ}$. While there are many Borel measures μ, the choice of Borel measure which assigns $\mu ([a,b])=b-a$ for every interval $[a,b]$ is sometimes called "the" Borel measure on $\mathbb{ℝ}$. In practice, even "the" Borel measure is not the most useful measure defined on the σ-algebra of Borel sets; indeed, the Lebesgue measure $\lambda$ is an extension of "the" Borel measure which possesses the crucial property that it is a complete measure (unlike the Borel measure). To clarify, when one says that the Lebesgue measure $\lambda$ is an extension of the Borel measure $\mu$, it means that every Borel-measurable set E is also a Lebesgue-measurable set, and the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., $\lambda (E)=\mu (E)$ for every Borel measurable set).

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Borel measure at Encyclopedia of Mathematics