456 APPENDIX A. GREEN’S THEOREM FOR A JORDAN CURVE

By the dominated convergence theorem,

limδ→0

∫Iδ(Qx−Py)dm2 =

∫Ui

(Qx−Py)dm2

where m2 denotes two dimensional Lebesgue measure discussed earlier. Let ∂R denotethe boundary of R for R one of these regions of Lemma A.0.1 oriented as described. Letwδ (R)

2 denote

(max{Q(x) : x ∈ ∂R}−min{Q(x) : x ∈ ∂R})2

+(max{P(x) : x ∈ ∂R}−min{P(x) : x ∈ ∂R})2

By uniform continuity of P,Q on the compact set Ui ∪ J, if δ is small enough, wδ (R) < ε

for all R ∈Bδ . Then for R ∈Bδ , it follows from Theorem 5.2.1, for |∂R| denoting thelength of ∂R, ∣∣∣∣∫

∂Rf ·dγ

∣∣∣∣≤ (12

)wδ (R)(|∂R|)< ε (|∂R|) (1.1)

whenever δ is small enough. Always let δ be this small.Also since the line integrals cancel on shared edges

∑R∈Iδ

∫∂R

f ·dγ + ∑R∈Bδ

∫∂R

f ·dγ =∫

Jf ·dγ (1.2)

Consider the second sum on the left. From 1.1,∣∣∣∣∣ ∑R∈Bδ

∫∂R

f ·dγ

∣∣∣∣∣≤ ∑R∈Bδ

∣∣∣∣∫∂R

f ·dγ

∣∣∣∣≤ ε ∑R∈Bδ

(|∂R|)

Denote by JR the part of J which is contained in R ∈Bδ . Then the above sum equals

ε

(∑

R∈Bδ

(|JR|+ |∂Rδ |)

)=

(ε |J|+ ε ∑

R∈Bδ

|∂Rδ |)

where |∂Rδ | is the sum of the lengths of the straight edges of Rδ . This last sum is easyto estimate. Recall from A.0.1 there are no more than 4+ 128|J|

δof these border regions.

Furthermore, the sum of the lengths of all four edges of one of these is no more than 4δ

and so

∑R∈Bδ

|∂Rδ | ≤ 4(

4+128 |J|

δ

)4δ = 1024 |J|+64δ .

Thus∣∣∣∑R∈Bδ

∫∂R f ·dγ

∣∣∣ ≤ ε (1025 |J|+64δ ) Let εn → 0 and let δ n be the correspondingsequence of δ such that δ n→ 0 also. Hence

limn→∞

∣∣∣∣∣∣ ∑R∈Bδn

∫∂R

f ·dγ

∣∣∣∣∣∣= 0.

Then using Green’s theorem proved above for squares,∫J

f ·dγ = limn→∞

∑R∈Iδn

∫∂R

f ·dγ + limn→∞

∑R∈Bδn

∫∂R

f ·dγ