Weak convergence in L ^p

The weak convergence in ${\displaystyle {\mathcal {L}}^{p}}$ and the weak convergence in ${\displaystyle L^{p}}$ are two closely related convergence terms for sequences of functions from measure theory . They are a special case of weak convergence in the sense of functional analysis for sequences in L ^p spaces . It should be noted that there are several different concepts of weak convergence in measure theory and stochastics , and these should not be confused with one another. In contrast to the weak convergence in or , the norm convergence, that is to say the convergence in the p-th mean, is then also referred to as strong convergence in or . ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle L^{p}}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle L^{p}}$

definition

Given is a measure space as well as and , thus with , the index to be conjugated . In addition, let out , in short , the space of p-fold integrable functions . The sequence of functions is said to be weakly convergent against if it holds for all that ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle p\in [1,\infty )}$ ${\displaystyle q:={\tfrac {1}{1-1/p}}}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$ ${\displaystyle {\tfrac {1}{\infty }}=0}$ ${\displaystyle p}$ ${\displaystyle f,(f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}^{p}(X,{\mathcal {A}},\mu )}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle g\in {\mathcal {L}}^{q}}$

${\displaystyle \lim _{n\to \infty }\int _{X}f_{n}g\mathrm {d} \mu =\int _{X}fg\mathrm {d} \mu }$

is. The weak convergence of functions is defined analogously . You then write in both cases . ${\displaystyle L^{p}}$ ${\displaystyle f_{n}\rightharpoonup f}$

classification

In functional analysis , weak convergence is understood to mean the following: starting from a normalized vector space , the topological dual space is formed ${\displaystyle V}$

${\displaystyle V':=\{T\;|\;T\colon V\to \mathbb {K} {\text{ ist linear und stetig }}\}}$.

A sequence in is then called weakly convergent to if ${\displaystyle (x_{n})_{n\in N}}$ ${\displaystyle V}$ ${\displaystyle x\in V}$

${\displaystyle \lim _{n\to \infty }T(x_{n})=T(x){\text{ für alle }}T\in V'}$

is. If we now consider as normed vector space to for , the dual space is norm isomorphic to (see also duality of Lp-spaces ), with the to conjugate Index is so . Every element from the dual space is then of the form ${\displaystyle {\mathcal {L}}^{p}(X,{\mathcal {A}},\mu )}$ ${\displaystyle p=(1,\infty )}$ ${\displaystyle {\mathcal {L}}^{q}(X,{\mathcal {A}},\mu )}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

${\displaystyle T_{g}(\cdot )=\int g\cdot \mathrm {d} \mu {\text{ für ein }}g\in {\mathcal {L}}^{q}}$.

Thus a sequence of weakly converges in if ${\displaystyle (f_{n})_{n\in \mathbb {N} }\in {\mathcal {L}}^{p}}$ ${\displaystyle L^{p}}$

${\displaystyle \lim _{n\to \infty }T_{g}(f_{n})=T_{g}(f)}$

for all that corresponds to the definition given above. The weak convergence in is thus a special case of the weak convergence in the sense of functional analysis and also a standard example of this. ${\displaystyle g\in {\mathcal {L}}^{q}}$ ${\displaystyle {\mathcal {L}}^{p}}$

Uniqueness

The limit of a weakly convergent sequence in only up to a - null set uniquely determined. This means that if the sequence of functions converges weakly against and weakly against, then - is almost everywhere . ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle \mu }$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f=g}$ ${\displaystyle \mu }$

Accordingly, the limit value in the case of the weak convergence is clearly determined due to the insensitivity to zero quantities. ${\displaystyle L^{p}}$

Relationship to other convergence terms

Customized local convergence

From the convergence locally made to measure , the weak convergence may follow. If a sequence from converges to locally according to measure and if the sequence of real numbers is bounded, then the sequence also converges weakly to . ${\displaystyle p\in (1,\infty )}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle f\in {\mathcal {L}}^{p}}$ ${\displaystyle (\|f_{n}\|_{p})_{n\in \mathbb {N} }}$ ${\displaystyle f}$

For this statement is generally not correct, as the following example shows: If one considers the measure space , then the sequence converges ${\displaystyle p=1}$ ${\displaystyle ([0,1],{\mathcal {B}}([0,1]),\lambda |_{[0,1]})}$

${\displaystyle f_{n}=n\chi _{[0,1/n]}}$

locally made to measure towards 0 and it's for everyone . But then for the constant function it is off ${\displaystyle \|f_{n}\|_{1}=1}$ ${\displaystyle n}$ ${\displaystyle g=1}$ ${\displaystyle {\mathcal {L}}^{\infty }}$

${\displaystyle \int _{X}f_{n}g\mathrm {d} \lambda =1}$.

Thus the sequence does not converge weakly to 0.

Convergence in the pth mean

Every sequence that converges in the p-th mean also converges for weak, because it follows from the Hölder inequality ${\displaystyle p\in [1,\infty )}$

${\displaystyle \left|\int _{X}f_{n}g\mathrm {d} \mu -\int _{X}fg\mathrm {d} \mu \right|\leq \|f_{n}-f\|_{p}\|g\|_{q}}$,

thus there is a convergent majorante. The limit values ​​then match. The Radon-Riesz also provides the reversal on one condition. It says that for a function sequence converges in the p-th mean if and only if it converges weakly and the sequence of the norms of the function sequence converges to the norm of the limit function. ${\displaystyle p\in (1,\infty )}$

literature

• Hans Wilhelm Alt: Linear Functional Analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , doi : 10.1007 / 978-3-642-22261-0 .
• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .