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Let f be a continuous function on [a,b]. Suppose that $\int_a^b f(t)d\alpha(t)=0$ for all $\alpha(t)$. What can we conclude about f?
The question has a hint: consider a piecewise constant function $\alpha(t)$. But I still have no idea to do this question.
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
== Formal definition ==
Let
X
{\displaystyle X}
be a locally compact Hausdorff space, and let
B
(
X
)
{\displaystyle {\mathfrak {B}}(X)}
be the smallest σ-algebra that contains the open sets of...
I'm currently reading Spanier's AT.
Theorem (Eilenberg-Zilber theorem) On the category of ordered pairs of topological spaces $X$ and $Y$, there is a natural chain equivalence of the functor $\Delta(X\times Y)$ with the functor $\Delta(X)\otimes \Delta(Y)$.
Proof outline: Both functors are free...
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