10.12. GOOD LAMBDA INEQUALITY 299
Note that the function t→ µ ([ f > t]) is a decreasing function. Therefore, one can makesense of an improper Riemann Stieltjes integral
∫∞
0 µ ([ f > t])dF (t) . With more work, onecan have this equal to the corresponding Lebesgue integral above.
10.12 Good Lambda InequalityThere is a very interesting and important inequality called the good lambda inequality (Iam not sure if there is a bad lambda inequality.) which follows from the above theory ofdistribution functions. It involves the inequality
µ ([ f > βλ ]∩ [g≤ δλ ])≤ φ (δ )µ ([ f > λ ]) (10.13)
for β > 1, nonnegative functions f ,g and is supposed to hold for all small positive δ andφ (δ )→ 0 as δ→ 0. Note the left side is small when g is large and f is small. The inequalityinvolves dominating an integral involving f with one involving g as described below. Asabove, ν is the Lebesgue Stieltjes measure described above in terms of F , an increasingfunction. Is there any way to see the inequality in 10.13 might make sense? Look at theexpression on the left. If δ is small enough, you might think that the intersection of the twosets would have smaller measure than µ ([ f > λ ]).
Theorem 10.12.1 Let (Ω,F ,µ) be a finite measure space and let F be a continu-ous increasing function defined on [0,∞) such that F (0) = 0. Suppose also that for everyα > 1, there exists a constant Cα such that for all x ∈ [0,∞),F (αx)≤Cα F (x) . Also sup-pose f ,g are nonnegative measurable functions and there exists β > 1, such that for allλ > 0 and 1 > δ > 0,
µ ([ f > βλ ]∩ [g≤ δλ ])≤ φ (δ )µ ([ f > λ ]) (10.14)
where limδ→0+ φ (δ ) = 0 and φ is increasing. Under these conditions, there exists a con-stant C depending only on β ,φ such that∫
Ω
F ( f (ω))dµ (ω)≤C∫
Ω
F (g(ω))dµ (ω) .
Proof: Let β > 1 be as given above. First suppose f is bounded. This is so there can beno question of existence of the integrals.
∫Ω
F ( f )dµ =∫
ΩF(
βfβ
)dµ ≤Cβ
∫Ω
F(
fβ
)dµ.
Let ν be the Lebesgue Stieltjes measure which comes from F , (dν = dF). From Theorem10.11.6, Cβ
∫Ω
F(
fβ
)dµ = Cβ
∫∞
0 µ ([ f/β > λ ])dν = Cβ
∫∞
0 µ ([ f > βλ ])dν . Now usingthe given inequality, ∫
Ω
F ( f )dµ =
Cβ
∫∞
0µ ([ f > βλ ]∩ [g≤ δλ ])dν (λ )+Cβ
∫∞
0µ ([ f > βλ ]∩ [g > δλ ])dν (λ )
≤ Cβ φ (δ )∫
∞
0µ ([ f > λ ])dν (λ )+Cβ
∫∞
0µ ([g > δλ ])dν (λ )
≤ Cβ φ (δ )∫
Ω
F ( f )dµ +Cβ
∫Ω
F( g
δ
)dµ