In order for convenience in checking definition along learning in Math, I found it necessary to gather definitions of the same kind so that distinguishing the differences among them will no longer be a tedious job. This work is very hard to do even many materials are available, if you want to use for commercial use, please contact me for permission. Furthermore, the work is still long from finished, if you want to contribute for more or better definitions, please contact me.
Tianyu Zhang
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Theorem 8.1
Result
If E is a set contains each Ei for 1≤ i ≤n, then the measure of E is greater or equal to the sum of all the measures of Ei. Note that this holds only for both E and Ei’s are bounded, left closed, and right open intervals. It may fail to be true in the more general case.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
Related Topics
Theorem 8.2
Statement
Result
The measure of a closed interval is strictly smaller than the sum of the measures of its finite cover generated by bounded open intervals.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
Related Topics
Theorem 8.3
Statement
Result
The measure is monotone.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
Related Topics
- Definitions
- Theorems/Corollary/Propositions/Lemmas
Theorem 8.4
Statement
Result
measure is a countably additive set function
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
Related Topics
- Definitions
- Additive(countably additive)
- Bounded
- Set Function
- Theorems/Corollary/Propositions/Lemmas
Theorem 8.5
Statement
Result
Existence of finite measure
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
Related Topics
- Definitions
- Theorems/Corollary/Propositions/Lemmas
Theorem 9.1
Theorem 9.2
Theorem 9.3
Theorem 9.4
Theorem 9.5
Theorem 9.6
Statement
Result
Construction from set function to measurer on a ring. If the set function on a ring is finite, nonnegative, additive, and continuous from above at 0 or continuous from below in this set, then it is sufficient to build a measure from such a set function.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 9
Related Topics
- Definitions
- Theorems/Corollary/Propositions/Lemmas
Theorem 10.1
Statement
Result
This theorem shows us how to build an outer measure on the hereditary sigma-ring from a measure on a ring. Indeed, the outer measure on a hereditary sigma-ring is an extension of the measure on the ring. Further, the total sigma-finite property is preserved in such an extension.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 10
Related Topics
- Definitions
- Theorems/Corollary/Propositions/Lemmas
Theorem 11.1
Statement
Result
This theorem provides a way to construct a ring from mu*-measurable sets on a hereditary sigma-ring. In the next Theorem, we make a further assumption that this construction of ring is in fact a Sigma-Ring.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
Related Topics
- Definitions
- Theorems/Corollary/Propositions/Lemmas
Theorem 11.2
Statement
Result
The first part of the theorem gives a further assumption to the previous theorem. While the second part declares that: if a set in sigma-ring could be expressed as a countable union of disjoint subsets in the sigma-ring, then the outer measure of the intersection between an arbitrary set in the sigma-ring H and such a set is in fact the countable sum of outer measure throughout each intersection between that arbitrary set and the disjoint sets.
Reference
Measure Theory, Paul R. Halmos, Chapter 2 Section 8
