62.5. SOME MAXIMAL ESTIMATES 2071

Let m→ ∞ in the above to obtain

P(∪t∈D [X (t)≥ λ ]) = P([

supt∈D

X (t)≥ λ

])≤ 1

λp

∫Ω

X[supt∈D X(t)≥λ ]X (T )p dP.

(62.5.20)From now on, assume that for a.e. ω ∈Ω, t→ X (t)(ω) is right continuous. Then with thisassumption, the following claim holds.

supt∈[S,T ]

X (t)≡ X∗ = supt∈D

X (t)

which verifies that X∗ is measurable. Then from 62.5.20,

P([X∗ ≥ λ ]) = P([

supt∈D

X (t)≥ λ

])≤ 1

λp

∫Ω

X[supt∈D X(t)≥λ ]X (T )p dP

=1

λp

∫Ω

X[X∗≥λ ]X (T )p dP

Now consider the other inequality. Using the distribution function technique and theabove estimate obtained in the first part,

E ((X∗)p) =∫

∞

0pα

p−1P([X∗ > α])dα

≤∫

∞

0pα

p−1P([X∗ ≥ α])dα

≤∫

∞

0pα

p−1 1α

∫Ω

X[X∗≥α]X (T )dPdα

= p∫

Ω

∫ X∗

0α

p−2dαX (T )dP

=p

p−1

∫Ω

(X∗)p−1 X (T )dP

≤ pp−1

(∫Ω

(X∗)p)1/p′(∫

Ω

X (T )p)1/p

=p

p−1E (X (T )p)

1/p E ((X∗)p)1/p′

.

Of course it would be nice to divide both sides by E ((X∗)p)1/p′ but we don’t know that

this is finite. One can use a stopped submartingale which will have X (t) bounded, divide,and then let the stopping time increase to ∞. However, this is discussed later.

With Theorem 62.5.2, here is an important maximal estimate for martingales havingvalues in E, a real separable Banach space.