8.5. TRANSLATION INVARIANCE LEBESGUE MEASURE 197

Theorem 8.4.3 Let f ≥ 0 and be Borel measurable. Then for any permutation(i1, · · · , ip) , ∫

f dmp =∫· · ·∫

f (x1, · · · ,xp)dm1 (xi1) · · ·dm1(xip

)(8.10)

Proof: By Theorem 6.1.10 there is an increasing sequence of simple, Borel measurablefunctions {sn} which converges pointwise to f . Since each is a finite linear combinationof indicator functions of Borel sets,∫

sndmp =∫· · ·∫

sn (x1, · · · ,xp)dm1 (xi1) · · ·dm1(xip

)Now apply the monotone convergence theorem to the succession of iterated integrals onthe right and to the single integral on the left to obtain 8.10. ■

Corollary 8.4.4 Suppose f ∈ L1 (Rp,mp) and f is Borel measurable. Then 8.10 holdsfor f .

Proof: This is obvious from applying Theorem 8.4.3 to the positive and negative partsof the real and imaginary parts of f . ■

Another thing should probably be noted. You can use Fubini’s theorem even if the func-tion is not Borel measurable. This depends on Corollary 8.2.11. Say f ∈ L1 (Rp,mp) so itis Lebesgue measurable but possibly not Borel measurable. Then from this corollary, thereis a set of measure zero N such that for x /∈ N, f (x) = g(x) where g is Borel measurable.By regularity, we can also assume N is Borel measurable. Then

∫f dmp =

∫XNC gdmp +

=0︷ ︸︸ ︷∫XN f dmp

=∫· · ·∫

XNC g(x1, · · · ,xp)dm1 (xi1) · · ·dm1(xip

)=

∫· · ·∫

g(x1, · · · ,xp)dm1 (xi1) · · ·dm1(xip

)Since g = f in L1 (Rp) , you can typically use g as a representative of f when using anysort of computation involving iterated integrals. The thing you want is

∫f dmp the iterated

integral is a tool for finding it. Therefore, no harm is done in using g rather than f .

8.5 Translation Invariance Lebesgue MeasureA very important property of Lebesgue measure is that it is translation invariant.

Definition 8.5.1 For E a set, E +x will be {y+x : y ∈ E} .

Theorem 8.5.2 Let E ∈Fp. Then mp (E) = mp (E + z) .

Proof: Let z = (z1, · · · ,zp) . The conclusion is obvious if E is an open rectangle

E =p

∏i=1

(ai,bi) .