Proposition 2 : Outer Measure is translation Invariant

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To prove this we need to know something
i.e. Length of Interval is translation invariant





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other related query
outermeasure is translate invartiant
prove that outer measure is translate invariant
proof of outer measurte istranlation invariant hindi
translation doesn't effect outermeasure


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Proposition 2 Outer measure is translation invariant, that is, for any set A and number y,
m*(A + y) = m*(A).
Proof Observe that if {Ik}k 1 is any countable collection of sets, then {Ik}k01 covers A if and
only if Ilk + y}k 1 covers A + y. Moreover, if each Ik is an open interval, then each Ik + y is
an open interval of the same length and so
00 00
I1(Ik)=I1(Ik+Y)-
k=1 k=1
The conclusion follows from these two observations.
Translation Invariance of the Lebesgue Outer Measure