1. Any measure defined on the Borel sets is called a Borel measure . 2. Without the condition of regularity the Borel measure need not be unique. 3. For some Borel measures \ mu _ i. 4. Furthermore, the Dirac delta function is not a function but it is a finite Borel measure . 5. If is a finite Borel measure on, then the Fourier Stieltjes transform of is the operator on defined by 6. For a Borel measure \ mu on a Euclidean space \ mathbb { R } ^ { n } define 7. By Carath�odory's extension theorem, there is a unique Borel measure on which agrees with on every interval. 8. In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on [ 0, " ). 9. This definition makes sense if " x " is an integrable function ( in distribution, or is a finite Borel measure . 10. The linear functional taking a continuous function to its value at ? corresponds to the regular Borel measure with a point mass at ?.
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