µ(A) + µ(B) = µ(A \ B + A )" B) + µ(B) = µ(A \ B) + µ(B) + µ(A )" B) =
= µ(A \ B + B) + µ(A )" B) = µ(A *" B) + µ(A )" B).
" µ( ) = = 0
" A1, A2, ... " F, Ai )" Aj = i = j

n An )" &!0 =

n An )" &!0 =

"n "n>n An )" &!0 = .
0 0
n > n0 µ(An) = 0
" "

( An) )" &!0 = (An )" &!0) = (An )" &!0) = ( An) )" &!0
n=1 n=1 nd"nO nd"n0
1 d" k d" n0 Ak )" &!0
" "

µ( An) = µ( An) = ( (An)"&!0)) = (An)"&!0) = µ(An) = µ(An) .
n=1 nd"n0 nd"n0 nd"n0 nd"n0 n=1
1 d" k d" n0 Ak )" &!0

µ(Ak) = +" ( An) )" &!0 = (An )" &!0)
nd"n0 nd"n0
" "

µ( An) = µ( An) = +" = µ(An) .
n=1 nd"n0 n=1
n An )" &!0 =

(kn) Ak )" &!0 =

n
" " "

( An) )" &!0 = (An )" &!0) ‡" (Ak )" &!0)
n
n=1 n=1 n=1
" " "

µ( An) = +" µ(An) e" µ(Ak ) = +" "nµ(Ak ) e" 1 .
n n
n=1 n=1 n=1
 µ Ô! µ(&!) < +" Ô! &! )" &!0 = &!0
Ò! µ Ã
"

"(A )Ä…"F&! = Aj µ(Aj) < +" j = 1, 2, ... .
j
j=1
j Aj )" &!0
" "

&!0 = &! )" &!0 = ( Aj) )" &!0 = (Aj )" &!0)
j=1 j=1
Ð! &!0 &!0 = {É1, É2, É3, ...}
A0 = &! \ &!0, A1 = {É1}, As = {É2}, ... .
"

&! = Aj µ(A1) = 0, µ(An) = 1 n e" 1 .
j=1
µ Ô! µ(&!) = 1 Ô! (&! )" &!0) = &!0 = 1
" Ä…( ) = 0 x0 "
/
" A1, A2, ... Ä…" X Ai )" Aj = i = j

" "

Ä…( ) = 0 Ô! x0 " Aj Ô! "j x0 " Aj Ô! "j Ä…(Aj) = 0
/ /
j=1 j=1
" "

Ä…( Aj) = 1 Ô! x0 " Aj Ô! "j x0 " Aj "j=j x0 " Aj Ô!
/

0 0 0
j=1 j=1
Ô! "j Ä…(Aj ) = 1 "j=j Ä…(Aj) = 0

0 0 0
" "
Ä…( Aj) = Ä…(Aj)
j=1 j=1
Ä… Ä…(X) = 1
" µ( ) = 0
" A1, A2, ... " B1 Ai )" Aj = i = j

"
i Ai (µ(Ai) = 0) Ai
i=1
" "

µ( Ai) = 0 = µ(Ai) .
i=1 i=1
"
i0 Ai (µ(Ai ) = +") Ai
0 0
i=1
" "

µ( Ai) = +" = µ(Ai) .
i=1 i=1
1 1 1 1 1
[0, 1] = {0} *" ( , 1] *" ( , ] *" ( , ] *" ... .
2 3 2 4 3
1 1
A1 = {0}, An = ( , ], n e" 2 .
n n - 1
1 1
µ(A1) = 1, µ(An) = ln - ln = ln nn - 1 < +" .
n - 1 n
" "

1 1 1 1
µ([0, 1]) = µ({0}) + µ(( , ]) = 1 + (ln - ln ) =
n n - 1 n - 1 n
n=2 n=2
1
= 1 + lim (- ln ) = 1 + lim (ln n) = +".
n" n"
n
" "
µ([1, 2]) = µ((-", 2] \ (-", 1))
"
an 2 an " Q+
"

"
µ((-", 2]) = µ( (-", aj]) = lim µ((-", aj]) = lim a2 = 2 .
j
j" j"
j=1
"

1 1 1
µ((-", 1)) = µ( (-", 1 - ]) = lim µ((-", 1 - ]) = lim (1 - )2 = 1 .
n" n"
n n n
n=1
" "
µ([1, 2) = µ((-", 2]) - µ((-", 1)) = 2 - 1 = 1 .

µ(A) = µ( (i, i + 1] × {0}) = µ((i, i + 1] × {0})
i"Z i"Z
i " Z µ((i, i + 1] × {0}) = 0
"

1 1
µ((i, i + 1] × {0}) = µ( (i, i + 1] × [0, ]) = lim µ((i, i + 1] × [0, ]) d"
j"
j j
j=1
1 1 1
d" lim µ([i, i + 1] × [0, ]) = lim (i + 1 - i) · = lim = 0.
j" j" j"
j j j