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Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions. Such proofs rely on the axiom of choice in an essential way, in the sense that Zermelo–Fraenkel set theory without the axiom of choice does not prove the existence of such functions.
where α(x) is a function of bounded variation on the interval [0, 1], and the integral is a Riemann–Stieltjes integral. Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation (that assigns to each function of bounded variation the corresponding Lebesgue–Stieltjes measure ...
Short-term memory has limited capacity and is often referred to as "working-memory", however these are not the same. Working memory involves a different part of the brain and allows you to manipulate it after initial storage. The information that travels from sensory memory to short-term memory must pass through the Attention gateway. The ...
The basic processes theory of memory development states that these memory processes underlie all cognition, as it holds that all more complex cognitive activities are built by combining these basic processes in different ways. [5] Thus, these memory basic processes can be seen as domain-general processes that can be applied across various domains.
Lebesgue's theory defines integrals for a class of functions called measurable functions. A real-valued function f on E is measurable if the pre-image of every interval of the form ( t , ∞) is in X :
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
Historically, questions regarding the functional architecture of the mind have been divided into two different theories of the nature of the faculties. The first can be characterized as a horizontal view because it refers to mental processes as if they are interactions between faculties such as memory, imagination, judgement, and perception, which are not domain specific (e.g., a judgement ...
Lashley coined the term equipotentiality to define the idea that if one part of the brain is damaged, other parts of the brain will carry out the memory functions for that damaged part. "The apparent capacity of any intact part of a functional brain to carry out… the [memory] functions which are lost by the destruction of [other parts]". [1]