Reading through a text book and it states that the collection of step functions form a linear space.
I know a step function has the form: $$f(x) = \sum_{i=1}^{n}c_i \cdot m(I_i)$$
But what exactly is a linear space?
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3That's another name for a vector space. – wjmolina Mar 17 '19 at 20:49
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https://en.wikipedia.org/wiki/Vector_space#Definition – Clement C. Mar 17 '19 at 20:49
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2If you add two step functions you get another step function, and so on. – kimchi lover Mar 17 '19 at 20:49
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As the comments state, one way is to interpret the space as a vector space.
Note that however there is kind-of two operations in play here.
For step functions $f(x) = \sum_{i=1}^n c_i \cdot m(I_i), g(x) = \sum_{j=1}^n d_i \cdot m(I_i)$, we have
$$f(x) + g(x)= \sum_{i=1}^n (c_i + d_i) \cdot m(I_i)$$
Similarly, however, we have another property of step functions that it is additive in the second part (the indicator function part).
Suppose that we have functions $f(x) = \sum_{i=1}^n c_i \cdot m(I_i)$, $g(x) = \sum_{i=1}^n d_i \cdot m(J_i)$. Vector operations still behave properly because if we have that $I_i \cap J_i = \emptyset$ for all $i$, we have addition simply. Else, we can define $c_i + d_i$ on $I_i \cap J_i$ and $c_i$ on $I_i - J_i$ and $d_i$ on $J_i - I_i$. This is how to show that it can be still expressed in step function form.
twnly
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