---- > [!definition] Definition. ([[typewriter sequence]]) > > > > The **typewriter sequence** on the [[interval]] $[0,1] \subset \mathbb{R}$ is defined as the sequence of functions $\big((1_{[0, k2^{-j}]})_{k=1}^{j}\big)_{j =1}^{\infty},$ > where the nested sequences are implicitly concatenated. The idea is to 'scan over' $[0,1]$ at increasing levels of granularity (unsurprisingly, there is a connection to [[wavelets]]) and concatenate successive scans together. > > ![[typewriter_sequence.gif]] > From https://math.stackexchange.com/questions/1412091/the-typewriter-sequence > > ![[typewriter_sequence_heatmap.png]] > > The typewriter sequence is notable as an example witnessing the [[convergence in measure]] does not imply [[sequence|convergence]] [[almost-everywhere]] (and in turn does not imply [[pointwise convergence]]). ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```