100 1. Measure theory (ii) fn converges in L∞ norm to zero if and only if An → 0 as n → ∞. (iii) fn converges almost uniformly to zero if and only if An → 0 as n → ∞, or μ(EN ∗ ) → 0 as N → ∞. (iv) fn converges pointwise to zero if and only if An → 0 as n → ∞, or ∞ N=1 EN ∗ = ∅. (v) fn converges pointwise almost everywhere to zero if and only if An → 0 as n → ∞, or ∞ N=1 EN ∗ is a null set. (vi) fn converges in measure to zero if and only if An → 0 as n → ∞, or μ(En) → 0 as n → ∞. (vii) fn converges in L1 norm if and only if Anμ(En) → 0 as n → ∞. To put it more informally: When the height goes to zero, then one has convergence to zero in all modes except possibly for L1 convergence, which requires that the product of the height and the width goes to zero. If instead, the height is bounded away from zero and the width is positive, then we never have uniform or L∞ convergence, but we have convergence in measure if the width goes to zero, we have almost uniform convergence if the tail support (which has larger measure than the width) has measure that goes to zero, we have pointwise almost everywhere convergence if the tail support shrinks to a null set, and pointwise convergence if the tail support shrinks to the empty set. It is instructive to compare this exercise with Exercise 1.5.2, or with the four examples given in the introduction. In particular: (i) In the escape to horizontal infinity scenario, the height and width do not shrink to zero, but the tail set shrinks to the empty set (while remaining of infinite measure throughout). (ii) In the escape to width infinity scenario, the height goes to zero, but the width (and tail support) go to infinity, causing the L1 norm to stay bounded away from zero. (iii) In the escape to vertical infinity, the height goes to infinity, but the width (and tail support) go to zero (or the empty set), causing the L1 norm to stay bounded away from zero. (iv) In the typewriter example, the width goes to zero, but the height and the tail support stay fixed (and thus bounded away from zero). Remark 1.5.8. The monotone convergence theorem (Theorem 1.4.43) can also be specialised to this case. Observe that the fn = An1En are monotone increasing if and only if An ≤ An+1 and En ⊂ En+1 for each n. In such cases, observe that the fn converge pointwise to f := A1E, where A := limn→∞ An and E := ∞ n=1 En. The monotone convergence theorem then asserts that

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