Show that the algebra built on top of the $B$-partition contains every $A_i$-set.
I'm trying to study measure theory, a subject I finished a few years ago, but would like to become a bit better at. I'm trying to do the problems in the book as I go along. However, I had problems understanding the question as well as the approach to this problem, hoping you could help me.
Let $(A_i)_{i in I}$ be a finite collection of subsets of $mathcal{X}$, and let $(B_J)_{J subset I}$ be the partition of $mathcal{X}$ constructed from the $A$-sets. Show that the algebra built on top of the $B$-partition contains every $A$-set.
Hint: For any $A$-set $A_i$ and any $B$-atom $B_J$ either $B_J subset A_i$ or $B_J cap A_i = emptyset$.
Obviously I somehow have to make use of the definition of an algebra and a partition, but I'm not exactly sure how.
Thanks in advance for any help.
measure-theory
asked Dec 9 at 19:09
Nicolai Kofoed
256
add a comment |
I'm trying to study measure theory, a subject I finished a few years ago, but would like to become a bit better at. I'm trying to do the problems in the book as I go along. However, I had problems understanding the question as well as the approach to this problem, hoping you could help me.
Let $(A_i)_{i in I}$ be a finite collection of subsets of $mathcal{X}$, and let $(B_J)_{J subset I}$ be the partition of $mathcal{X}$ constructed from the $A$-sets. Show that the algebra built on top of the $B$-partition contains every $A$-set.
Hint: For any $A$-set $A_i$ and any $B$-atom $B_J$ either $B_J subset A_i$ or $B_J cap A_i = emptyset$.
Obviously I somehow have to make use of the definition of an algebra and a partition, but I'm not exactly sure how.
Thanks in advance for any help.
measure-theory
asked Dec 9 at 19:09
Nicolai Kofoed
256
add a comment |
I'm trying to study measure theory, a subject I finished a few years ago, but would like to become a bit better at. I'm trying to do the problems in the book as I go along. However, I had problems understanding the question as well as the approach to this problem, hoping you could help me.
Let $(A_i)_{i in I}$ be a finite collection of subsets of $mathcal{X}$, and let $(B_J)_{J subset I}$ be the partition of $mathcal{X}$ constructed from the $A$-sets. Show that the algebra built on top of the $B$-partition contains every $A$-set.
Hint: For any $A$-set $A_i$ and any $B$-atom $B_J$ either $B_J subset A_i$ or $B_J cap A_i = emptyset$.
Obviously I somehow have to make use of the definition of an algebra and a partition, but I'm not exactly sure how.
Thanks in advance for any help.
measure-theory
asked Dec 9 at 19:09
Nicolai Kofoed
256
I'm trying to study measure theory, a subject I finished a few years ago, but would like to become a bit better at. I'm trying to do the problems in the book as I go along. However, I had problems understanding the question as well as the approach to this problem, hoping you could help me.
Let $(A_i)_{i in I}$ be a finite collection of subsets of $mathcal{X}$, and let $(B_J)_{J subset I}$ be the partition of $mathcal{X}$ constructed from the $A$-sets. Show that the algebra built on top of the $B$-partition contains every $A$-set.
Hint: For any $A$-set $A_i$ and any $B$-atom $B_J$ either $B_J subset A_i$ or $B_J cap A_i = emptyset$.
Obviously I somehow have to make use of the definition of an algebra and a partition, but I'm not exactly sure how.
Thanks in advance for any help.
measure-theory
measure-theory
asked Dec 9 at 19:09
Nicolai Kofoed
256
asked Dec 9 at 19:09
Nicolai Kofoed
256
asked Dec 9 at 19:09
Nicolai Kofoed
256
asked Dec 9 at 19:09
Nicolai Kofoed
256
asked Dec 9 at 19:09
Nicolai Kofoed
256
256
add a comment |
add a comment |
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