How to prove that function is quasimetric
We introduce a non-negative function on $mathcal F$ x $mathcal F$ ($sigma$-algebra) by the rule:
$$d(A,B)= frac {P(A backslash B) + P(B backslash A)} {P(Acup B)}$$
in the case of $P (A ∪ B)> 0$ and $0$ otherwise. How to show that $d(A,B)$ is a quasimetric:
$$1. d(A,B) geq 0 $$
$$2. d(A,B) = d(B,A) $$
$$3. d(A,C) ≤ d(A,B) + d(B,C).$$ $$forall A, B, C in mathcal F ?$$
Properties $1$ and $2$ are clear but i don't know how to prove property $3$
probability-theory measure-theory
asked Nov 29 at 16:26
Lisa
265
add a comment |
We introduce a non-negative function on $mathcal F$ x $mathcal F$ ($sigma$-algebra) by the rule:
$$d(A,B)= frac {P(A backslash B) + P(B backslash A)} {P(Acup B)}$$
in the case of $P (A ∪ B)> 0$ and $0$ otherwise. How to show that $d(A,B)$ is a quasimetric:
$$1. d(A,B) geq 0 $$
$$2. d(A,B) = d(B,A) $$
$$3. d(A,C) ≤ d(A,B) + d(B,C).$$ $$forall A, B, C in mathcal F ?$$
Properties $1$ and $2$ are clear but i don't know how to prove property $3$
probability-theory measure-theory
asked Nov 29 at 16:26
Lisa
265
See section 1.2 in matwbn.icm.edu.pl/ksiazki/cm/cm6/cm6141.pdf
– d.k.o.
Nov 29 at 18:41
add a comment |
We introduce a non-negative function on $mathcal F$ x $mathcal F$ ($sigma$-algebra) by the rule:
$$d(A,B)= frac {P(A backslash B) + P(B backslash A)} {P(Acup B)}$$
in the case of $P (A ∪ B)> 0$ and $0$ otherwise. How to show that $d(A,B)$ is a quasimetric:
$$1. d(A,B) geq 0 $$
$$2. d(A,B) = d(B,A) $$
$$3. d(A,C) ≤ d(A,B) + d(B,C).$$ $$forall A, B, C in mathcal F ?$$
Properties $1$ and $2$ are clear but i don't know how to prove property $3$
probability-theory measure-theory
asked Nov 29 at 16:26
Lisa
265
We introduce a non-negative function on $mathcal F$ x $mathcal F$ ($sigma$-algebra) by the rule:
$$d(A,B)= frac {P(A backslash B) + P(B backslash A)} {P(Acup B)}$$
in the case of $P (A ∪ B)> 0$ and $0$ otherwise. How to show that $d(A,B)$ is a quasimetric:
$$1. d(A,B) geq 0 $$
$$2. d(A,B) = d(B,A) $$
$$3. d(A,C) ≤ d(A,B) + d(B,C).$$ $$forall A, B, C in mathcal F ?$$
Properties $1$ and $2$ are clear but i don't know how to prove property $3$
probability-theory measure-theory
probability-theory measure-theory
asked Nov 29 at 16:26
Lisa
265
asked Nov 29 at 16:26
Lisa
265
asked Nov 29 at 16:26
Lisa
265
asked Nov 29 at 16:26
Lisa
265
asked Nov 29 at 16:26
Lisa
265
265
See section 1.2 in matwbn.icm.edu.pl/ksiazki/cm/cm6/cm6141.pdf
– d.k.o.
Nov 29 at 18:41
add a comment |
See section 1.2 in matwbn.icm.edu.pl/ksiazki/cm/cm6/cm6141.pdf
– d.k.o.
Nov 29 at 18:41
See section 1.2 in matwbn.icm.edu.pl/ksiazki/cm/cm6/cm6141.pdf
– d.k.o.
Nov 29 at 18:41
See section 1.2 in matwbn.icm.edu.pl/ksiazki/cm/cm6/cm6141.pdf
– d.k.o.
Nov 29 at 18:41
add a comment |
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See section 1.2 in matwbn.icm.edu.pl/ksiazki/cm/cm6/cm6141.pdf
– d.k.o.
Nov 29 at 18:41