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&
/, 06(73 * .. 1 μ R .. 1
R 8 $ =
2!3 (∀x, y ∈ R) (μ(x − y) ≥ min{μ(x), μ(y)}),
2+3 (∀x, y ∈ R) (μ(xy) ≥ min{μ(x), μ(y)}),
/, 06(73 * .. 1 μ R .. 2&
3 R 8 2!3 2(3 (∀x, y ∈ R) (μ(xy) ≥ μ(y)) 2& μ(xy) ≥ μ(x)3&
4
μ 1 .. .. R, μ 1 .. R.
0 .. 1 μ X t ∈ [0, 1] U(μ; t) = {x ∈
X | μ(x) ≥ t} 1 μ. * .. 1 μ X μ(y) :=
t ∈ (0, 1] if y = x,
0
if y = x,
1 .. $ x t 1 [x; t].
0 .. 1 μ X, $ .. [x; t] 2,3 μ, 1 [x; t] ∈ μ, 26,73 μ(x) ≥ t.
2>3 ; $ μ, 1 [x; t] q μ, 26,73 μ(x) + t > 1.
0 .. [x; t] .. 1 μ X, $ 23
2 3
+ >
[x; t] ∈ μ [x; t] q μ.
[x; t] α μ [x; t] α μ α ∈ {∈, q, ∈ ∨ q }.
/, 06!73 * .. 1 μ R (∈, ∈ ∨ q )
.. 1 R x, y ∈ R t, r ∈ (0, 1],
2/3 [x; t] ∈ μ, [y; r] ∈ μ ⇒ [x + y; min{t, r}] ∈ ∨ q μ,
2)3 [x; t] ∈ μ ⇒ [−x; t] ∈ ∨ q μ,
2!3 [x; t] ∈ μ, [y; r] ∈ μ ⇒ [xy; min{t, r}] ∈ ∨ q μ.
/, 1 06!73 * .. 1 μ R (∈, ∈ ∨ q )
.. R 2!!3 μ (∈, ∈ ∨ q )
.. 1 R,
2!+3 (∀x, y ∈ R) (∀t ∈ (0, 1]) ([x; t] ∈ μ ⇒ [xy; t] ∈ ∨ q μ, [yx; t] ∈ ∨ q μ).
4 $ $ R k 1 [0, 1)
$ 8& 0 .. [x; t] .. 1 μ R,
$ 21!3 [x; t] qk μ μ(x) + t + k > 1.
21+3 [x; t] ∈ ∨ qk μ [x; t] ∈ μ [x; t] qk μ.
21(3 [x; t] α μ [x; t] α μ α ∈ { qk , ∈ ∨ qk }.
/, 2 06+73 * .. 1 μ R (∈, ∈ ∨ qk )
..
1 R x, y ∈ R t, r ∈ (0, 1],
21,3 [x; t] ∈ μ, [y; r] ∈ μ ⇒ [x + y; min{t, r}] ∈ ∨ qk μ,
21>3 [x; t] ∈ μ ⇒ [−x; t] ∈ ∨ qk μ,
213 [x; t] ∈ μ, [y; r] ∈ μ ⇒ [xy; min{t, r}] ∈ ∨ qk μ.
# 06+73 21,3 (∀x, y ∈ R) (μ(x + y) ≥ min{μ(x), μ(y), 1−k
}),
2+&!3
2
21>3 (∀x ∈ R) (μ(−x) ≥ min{μ(x), 1−k
}).
2+&+3
2
/, 06+73 * .. 1 μ R (∈, ∈ ∨ qk )
..
R (∈, ∈ ∨ qk )
.. 1 R $
[x; t] ∈ ∨ q μ
+ 21 3 (∀x, y ∈ R) (∀t ∈ (0, 1]) ([x; t] ∈ μ ⇒ [xy; t] ∈ ∨ qk μ, [yx; t] ∈ ∨ qk μ).
4 # 21 3 ; $
(∀x, y ∈ R) (min{μ(xy), μ(yx)} ≥ min{μ(x), 1−k
}).
2+&(3
2
' $ $ . (∈, ∈ ∨ qk )
.. R.
3
' 06+73 μ R (∈, ∈ ∨ qk )
R
2+&(3 (∀x, y ∈ R) (μ(x − y) ≥ min{μ(x), μ(y), 1−k
}).
2+&,3
2
(∈, ∈ ∨ qk )
/, ? μ 1 (∈, ∈ ∨ q )
.. R x ∈ R. -
k
.. 1 μx R 8 1 μx(u) = min{μ(u − x), 1−k
} u ∈ R 2
(∈, ∈ ∨ qk )
.. 1 x μ.
- (∈, ∈ ∨ qk )
.. 1 x μ $ k = 0 (∈, ∈ ∨ q )
.. 1 x μ.
1 Rµ∗ 2& Rµ 3 (∈, ∈ ∨ qk )
.. 2& (∈,
∈ ∨ q )
.. 3 μ R.
3
' μ (∈, ∈ ∨ q )
R. R k
(+) (·) μx + μy = μx+y μx · μy = μxy μx , μy ∈ Rµ∗ .
? a, b, c, d ∈ R 1 μa = μb μc = μd . -
= min μ(r − b), 1−k
min μ(r − a), 1−k
2
2
= min μ(r − d), 1−k
min μ(r − c), 1−k
2
2
∗
µ
2(&!3
2(&+3
r ∈ R. 4
$ # r = a r = c 2(&!3 2(&+3 1−k
= min μ(0), 1−k
= 2
min μ(a − b), 1−k
2(&(3
2
2
1−k
1−k
=
min
μ(0),
= 2 .
min μ(c − d), 1−k
2(&,3
2
2
+
1−k
= min
r =
a
+
c
−
d
2(&!3
min
μ(a
+
c
−
d
−
b),
2
1−k
1−k
1−k
μ(c − d), 2 = 2 μ(a + c − d − b) ≥ 2 . 0 r ∈ R, $ (μa + μc )(r) = μa+c (r) = min μ(r − (a + c)), 1−k
2
= min μ((r − b − d) − (a + c − b − d)), 1−k
2
≥ min μ(r − b − d), μ(a + c − b − d), 1−k
2
1−k
= min μ(r − b − d), 2 = μb+d (r) = (μb + μd )(r).
4
$ #
' μa + μc ≥ μb + μd . @ μb + μd ≥ μa + μc, μa + μc = μb + μd.
-
A +B $ 8& $ r ∈ R, $ (μa · μc )(r) = μac (r) = min μ(r − ac), 1−k
2
= min μ((r − bd) − (ac − bd)), 1−k
2
≥ min μ(r − bd), μ(ac − bd), 1−k
2
= min μ(r − bd), μ((a − b)c − b(d − c)), 1−k
2
≥ min μ(r − bd), μ((a − b)c), μ(b(d − c)), 1−k
2
≥ min μ(r − bd), μ(a − b), μ(c − d), 1−k
2
1−k
= min μ(r − bd), 2 = μbd (r) = (μb · μd )(r).
' μa · μc ≥ μb · μd . @ μb · μd ≥ μa · μc . -
μa · μc = μb · μd,
A ·B $ 8& 4 $ Rµ∗
$ μ0 μ−x μx.
+''" 06!73 μ (∈, ∈ ∨ q )
R. Rµ (+) (·) μx + μy = μx+y μx · μy = μxy μx , μy ∈ Rµ .
3
' 1∗ ! (∈, ∈ ∨ qk )
∗ μ R, μ Rµ μ(μx ) = μ(x) μx ∈ Rµ x ∈ R. μ (∈, ∈ ∨ qk )
Rµ∗ .
? x,y ∈ R. - μ(μx−μy ) = μ(μx−y ) = μ(x−y) ≥ min {μ(x), μ(y),
= min μ(μx ), μ(μy ), 1−k
min
x ·μy ), μ(μy · μx )} = min
2
{μ(μ
{μ(μxy ),
1−k
1−k
μ(μyx )} = min {μ(xy), μ(yx)} ≥ min μ(x), 2 = min μ(μx ), 2 .
- +&/ μ (∈, ∈ ∨ qk )
.. Rµ∗ .
1−k
2
+''" 2 06!73 ! (∈, ∈ ∨ q )
μ R, μ Rµ μ(μx ) = μ(x) μx ∈ Rµ x ∈ R. μ (∈, ∈ ∨ q )
Rµ .
# ! (∈, ∈ ∨ qk )
μ R,1−k
A := {x ∈ R | μx ≥ μ0}
B := {x ∈ R | μx = μ0} . A = B = U μ; 2 .
+ /
4
x ∈ A, μx (r) ≥ μ0(r) r ∈ R, μ(x) ≥ 1−k
1 #
2
r = 0. 4 $ μ0 (r) = min
μ(r), 1−k
= min μ(r − x + x), 1−k
≥
2
2
1−k
1−k
= min μ(r − x), 2
= μx (r) r ∈ R, min μ(r − x), μ(x), 2
μ0 ≥ μx. ' μ0 = μx, && x ∈ B. - A = B. 9
x ∈ A 1−k
μx (r) ≥ μ0 (r) μ(−x) ≥ min μ(0),
. * μ(x) ≥
2
r ∈ R. -1−k
1−k
1−k
min μ(−x),
2 . ' μ(x) ≥
21−k, x ∈ U μ;1−k2 . - $ .
A ⊆ U μ; 1−k
$
x
∈
U
- μ(x)≥ 2 , $ 2
μ; 2 . 1−k
= min
μ(r),1−k
μ(r − x) ≥ min
r ∈ R. -
2
2
μ(r), μ(x),
1−k
≥ min μ(r), 1−k
=
μ
μx (r) = min μ(r − x), 2
r ∈ R, &&
0 (r)
2
1−k
1−k
μx ≥ μ0 . - U μ; 2 ⊆ A, ; A = U μ; 2 .
+''" 06!73 ! (∈, ∈ ∨ q )
μ R, A = {x ∈ R |
B = {x ∈ R | μx = μ0} . A = B = U (μ; 0.5) .
3
' " μ (∈, ∈ ∨ qk )
R, μx ≥ μ0 }
f : R → Rµ∗ , x → μx
.
ker(f ) = U μ; 1−k
2
0 x, y ∈ R, $ f (x + y) = μx+y = μx + μy = f (x) +
f (y) f (xy) = μxy = μx · μy = f (x) · f (y). ' f &
$ Ker(f ) = {x ∈ R | f (x) = f (0)} = {x ∈ R | μx = μ0} = U μ; 1−k
1
2
? (&&
f - (&/ & ' 8
Rµ∗ .
R/U μ; 1−k
2
+''" 4 06!73 " μ (∈, ∈ ∨ q )
R, f : R → Rµ , x → μx
ker(f ) = U (μ; 0.5) . # R/U (μ; 0.5) Rµ .
3
' μ μ (∈, ∈ ∨ qk )
R Rµ∗ , $ ν R ν(x) = μ(μx ) x ∈ R. ν (∈, ∈ ∨ qk )
R.
0 x, y ∈ R, $ min
= min
{μ(μxy), μ(μyx )} =
{ν(xy), ν(yx)}
1−k
= min
ν(x),
min {μ(μx · μy ), μ(μy · μx )} ≥ min μ(μx ), 1−k
ν(x−
2
2
1−k
=
min
ν(x),
ν(y),
.
y) = μ(μx−y ) = μ(μx −μy ) ≥ min μ(μx ), μ(μy ), 1−k
2
2
- +&/ $ ν (∈, ∈ ∨ qk )
.. R.
+''" 06!73 μ μ (∈, ∈ ∨ q )
R R ,
µ
$ ν R ν(x) = μ(μx) x ∈ R. ν (∈, ∈ ∨ q )
R.
+ )
(∈, ∈ ∨ qk )
/, 1 ? μ 1 (∈, ∈ ∨ q )
.. R.
k
- .. 1 Radμ R 8 1
Radμ(x) :=
min sup{μ(xn ) | n ∈ N}, 1−k
if μ(x) <
2
μ(x)
if μ(x) ≥
1−k
,
2
1−k
2
2,&!3
(∈, ∈ ∨ qk )
.. μ.
- (∈, ∈ ∨ qk )
.. μ $ k = 0 (∈, ∈ ∨ q )
..
μ 2 6!73&
# 1 " μ (∈, ∈ ∨ q )
R, k
.
2!3 (∀m ∈ N) (∀x ∈ R) μ(mx) ≥ min μ(x), 1−k
2
.
2+3 (∀m, n ∈ N) (∀x ∈ R) m ≥ n ⇒ μ(xm) ≥ min μ(xn), 1−k
2
2!3 2+&!3 $ μ(2x) ≥ min μ(x), 1−k
x ∈ R.
2
- 2!3 2!3 m = r. -
m = 2. *
1−k
≥
min
μ(x),
. - 1 μ ((r + 1)x) ≥ min μ(rx), μ(x), 1−k
2
2
4 2!3 &
m, n ∈ N 1 m ≥ n. - μ (xm ) = μ (xm−n xn ) ≥
2+3
?
x ∈ R.
min μ (xn ) , 1−k
2
4
$ # k = 0 ? ,&+ $ $ &
+''" 1 06!73 " μ (∈, ∈ ∨ q )
R,
2!3
2+3
(∀m ∈ N) (∀x ∈ R) (μ(mx) ≥ min {μ(x), 0.5}) .
(∀m, n ∈ N) (∀x ∈ R) (m ≥ n ⇒ μ(xm ) ≥ min {μ(xn ), 0.5}) .
3
' 11 " μ (∈, ∈ ∨ q )
R,
k
(∈, ∈ ∨ qk )
(∈, ∈ ∨ qk )
R.
0 x, y ∈ R, μ(x − y) < 1−k
μ(x − y) ≥ 1−k
. *
2
2
μ(x − y) < 1−k
. @ R $ (x − y)m+n = axm + by n
2
$ a, b ∈ R m, n ∈ N. -
Radμ(x − y) = min sup {μ((x − y)r ) | r ∈ N} , 1−k
2
≥ sup min μ((x − y)r ), 1−k
|r∈N
2
m
n 1−k
≥ min μ((x − y)m+n), 1−k
+
by
),
=
min
μ(ax
2
2
m
n 1−k
≥
min
μ(x
.
≥ min μ(axm ), μ(by n), 1−k
),
μ(y
),
2
2
, $ $ =
@ μ(x − y) < 1−k
2
2,&+3
+/
23 μ(x) < 1−k
μ(y) < 1−k
,
2
2
23 μ(x) ≥ 1−k
μ(y) < 1−k
,
2
2
23 μ(x) < 1−k
μ(y) ≥ 1−k
.
2
2
0 8 $ 2,&+3 ,
Radμ(x − y) ≥ min min sup {μ(xm ) | m ∈ N} , 1−k
2
1−k
, 2
min sup {μ(y n ) | n ∈ N} , 1−k
2
= min Radμ(x), Radμ(y), 1−k
.
2
- Radμ(x) = μ(x). 2,&+3 $ 1−k , 2
Radμ(x − y) ≥ min μ(x), min sup {μ(y n) | n ∈ N} , 1−k
2
1−k
= min Radμ(x), Radμ(y), 2 .
- & @ μ(x − y) ≥ 1−k
.
2
1−k
1−k
- μ(x) < 1−k
μ(y)
<
.
'
Radμ(x
−
y)
=
μ(x
−
y)
≥
≥
2
2
2
. $ μ(xy) < 1−k
min Radμ(x), Radμ(y), 1−k
2
2
Radµ(xy) = min sup {µ((xy)n ) | n ∈ N} , 1−k
2
≥ min sup min µ(xn ), 1−k
| n ∈ N , 1−k
= min sup {µ(xn y n ) | n ∈ N} , 1−k
2
2
2
1−k , 2 = min Radµ(x), 1−k
.
= min min sup {µ(xn ) | n ∈ N} , 1−k
2
2
1−k
µ(xy) ≥ 1−k
2 . Radµ(xy) = µ(xy) ≥ 2 ≥ min {Radµ(x),
1−k
. Radµ (∈, ∈ ∨ qk ) R.
2
+''" 12 06!73 " μ (∈, ∈ ∨ q )
R, (∈, ∈ ∨ q )
(∈, ∈ ∨ q )
R.
6!7 @& "& 5# C& 0.. 1 8 !
%
% 2!))3 (/(D()(&
6+7 E& 5& : & *& F.G# & '& H .. 1
# 213&
6(7 <& :& ? 0.. 1 .. !
% % 2!)/+3 !((!()&
6,7 C& & C E& & ? 0.. 4 1 ..
@ & # 2!)/3
> !D>))&
56- "