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What is a disjoint sequence of sets?

{A[n]} sequence of sets

for all natural m,n, m!=n

A[m] and A[n] are disjoint

for all natural m,n, m!=n

A[m] and A[n] are disjoint

Prove that an increasing sequence of sets converges.

The limit is equal to the union of all the sets from this
sequence.

Prove that a decreasing sequence of sets converges.

The limit is equal to the intersection of all the sets from
this sequence.

Prove that a disjoint sequence of sets converges.

The limit is the empty set.

Let A,B c X.

Let 1(A) denote the characteristic function of the set A.

Prove that the two functions are equal:

1(A+B) = ???

Let 1(A) denote the characteristic function of the set A.

Prove that the two functions are equal:

1(A+B) = ???

1(A+B) = |1(A) - 1(B)|

Let A[n] c X for all natural n.

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

/\x:-X 1(lim_inf A[n])(x) = ???

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

/\x:-X 1(lim_inf A[n])(x) = ???

/\x:-X 1(lim_inf A[n])(x) = lim_inf 1(A[n])(x)

Let A[n] c X for all natural n. Let AcX.

If HcX, then let 1(H) denote the characteristic function of the set H defined on X.

(1) the sequence of sets {A[n]} converges to the set A

(2) the function 1(A[n]) converges pointwise to the function 1(A)

If HcX, then let 1(H) denote the characteristic function of the set H defined on X.

(1) the sequence of sets {A[n]} converges to the set A

(2) the function 1(A[n]) converges pointwise to the function 1(A)

page 13

first measure theory notebook

first measure theory notebook

(A\B) u (B\A) = ???

Express differently.

Express differently.

(A\B) u (B\A) = (A u B) \ (A n B)

(A u B) \ (A n B) = ???

Express differently.

Express differently.

(A u B) \ (A n B) = (A\B) u (B\A)

Is it true that

(A + B) c (A + C) u (C + B)

?

(A + B) c (A + C) u (C + B)

?

Yes.