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Miklós Schweitzer
1955 Miklós Schweitzer
2
2
Part of
1955 Miklós Schweitzer
Problems
(1)
Miklós Schweitzer 1955- Problem 2
Source:
9/30/2015
2. Let
f
1
(
x
)
,
…
,
f
n
(
x
)
f_{1}(x), \dots , f_{n}(x)
f
1
(
x
)
,
…
,
f
n
(
x
)
be Lebesgue integrable functions on
[
0
,
1
]
[0,1]
[
0
,
1
]
, with
∫
0
1
f
1
(
x
)
d
x
=
0
\int_{0}^{1}f_{1}(x) dx= 0
∫
0
1
f
1
(
x
)
d
x
=
0
(
i
=
1
,
…
,
n
)
(i=1,\dots ,n)
(
i
=
1
,
…
,
n
)
. Show that, for every
α
∈
(
0
,
1
)
\alpha \in (0,1)
α
∈
(
0
,
1
)
, there existis a subset
E
E
E
of
[
0
,
1
]
[0,1]
[
0
,
1
]
with measure
α
\alpha
α
, such that
∫
E
f
i
(
x
)
d
x
=
0
\int_{E}f_{i}(x)dx=0
∫
E
f
i
(
x
)
d
x
=
0
. (R. 17)
real analysis
function
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