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# Different Methods of Convergence

DOI link for Different Methods of Convergence

Different Methods of Convergence book

# Different Methods of Convergence

DOI link for Different Methods of Convergence

Different Methods of Convergence book

## ABSTRACT

Let
(
X
,
X
,
μ
)
$ (X, {\mathcal{X}},\text{ }\mu ) $
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be a measure space. In this book we have treated four different types of convergence of a sequence of {f
_{
j
}} functions to a limit function f:

pointwise convergence: For each ɛ > 0 and each x ∊ X there is a number J > 0 such that, if j > J, then |f
_{
j
}(x) - f(x)| < ɛ.

uniform convergence: For each ɛ > 0 there is a number J > 0 such that, if j > J and x ∊ X, then |f
_{
j
}(x) - f(x)| < ɛ.

convergence almost everywhere: There exists a set E ⊆ X with μ(E) = 0 so that, for every ɛ > 0, and each
x
∈
X
\
E
$ x \in X\backslash E $
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, there is a number J > 0 such that, for j > J, we have |f
_{
j
}(x) - f(x)| < ɛ.

convergence in L
^{
p
}, 1 ≤ p < ∞: For each ɛ > 0 there is a number J > 0 so that, if j > J, then