Is the following "definition" for an improper integral of a function with two critical endpoints equivalent to the "standard" one?

Question

The following are standard definitions.

A function $f: [a,b) \rightarrow E$ (where $a < b \leq \infty$ and $E$ is a real Banach space) is called improperly Riemann integrable if it is Riemann integrable on $[a,b']$ for all $b' \in [a,b)$ and $\lim_{b' \uparrow b}\int_a^{b'}f(x)dx$ exists. In that case, the improper Riemann integral is $\int_a^bf(x)dx := \lim_{b' \uparrow b}\int_a^{b'}f(x)dx$

Analogously one defines the improper integral for functions $f: (a,b] \rightarrow E$ (where $-\infty \leq a < b$).

The improper integral of a function $f: (a,b) \rightarrow E$ (where $-\infty \leq a < b \leq \infty$) is $$\int_a^bf(x)dx = \int_a^cf(x)dx +\int_c^bf(x)dx,$$ if both improper integrals on the right exist for some (and therefore any) $c \in (a,b)$.

What about the following alternative definition?

A function $f: (a,b) \rightarrow E$ is called improperly Riemann integrable if it is Riemann integrable on $[a',b']$ for all $a',b' \in (a,b)$ and $\lim_{(a',b') \rightarrow (a,b)}\int_{a'}^{b'}f(x)dx$ exists. In that case, the improper integral is $$\int_a^bf(x)dx = \lim_{(a',b') \rightarrow (a,b)}\int_{a'}^{b'}f(x)dx$$.

Question: Are these two definitions for improper Riemann integrability of a function $f: (a,b) \rightarrow E$ equivalent?

I proved that the standard definition implies the alternative one and that if both hold, they yield the same value for the improper integral. However, I cannot prove or give a counterexample to the statement that the alternative definition implies the standard one. Either I am completely missing something, or this is actually an interesting question.

Answer 1

First, recall the "Cauchy criterion for limits". Namely, let $F : (a, b] \to \Bbb R$ be a function.
Then, $\lim_{x \downarrow a} F(x)$ exists iff for every $\epsilon > 0$, there exists a (relatively) open neighbourhood $U \subset [a, b]$ of $a$ such that $|F(x) - F(y)| < \epsilon$ for all $x, y \in U \setminus \{a\} = \dot U$.

(The reason for writing in terms of open neighbourhoods is that it takes care of $a = -\infty$ and $a \in \Bbb R$ both at once.)

Similarly, we have a similar criterion for functions on $[a, b)$ with a limit at $b$.

(The motivation for considering this is mentioned at the end.)

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$\newcommand{\md}[1]{{\left\lvert #1 \right\lvert}}$ Assume that $f$ is improperly Riemann integrable on $(a, b)$ as per the alternate definition. Let $L$ be the value of the integral.

Fix any $c \in (a, b)$.

We show that both the desired integrals exist by showing that the Cauchy criterion is satisfied. That is, we wish to show that $$\lim_{a' \downarrow a} \int_{a'}^{c} f \quad\text{and}\quad \lim_{b' \uparrow b} \int_{c}^{b'} f$$ exist.

To this end, let $\epsilon > 0$ be given.
By the existence of limit as per alternate definition, there exist appropriate open neighbourhoods $U_a$ and $U_b$ of $a$ and $b$, respectively such that $$\left\lvert\int_{a'}^{b'} f - L\right\rvert < \frac{\epsilon}{2}$$ for all $(a', b') \in \dot U_a \times \dot U_b \subset (a, b) \times (a, b)$.
(The dot on top indicates that we are deleting the points $a$ and $b$.)

Now, fix $a' \in \dot U_a$ and let $b', b'' \in \dot U_b$ be arbitrary. Then, we have

\begin{align} \md{\int_c^{b'}f - \int_c^{b''}f} &= \md{\int_{a'}^{b'}f - \int_{a'}^{b''}f} \\ &\le \md{\int_{a'}^{b'}f - L} + \md{\int_{a'}^{b''}f - L} \\ &< \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon. \end{align}

Thus, we see that $$\lim_{b' \uparrow b} \int_{c}^{b'}f$$ exists. Similarly, the other desired limit also exists.

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The motivation for using the Cauchy criterion: We wanted to show that the two limits exist. However, we didn't have any guess limit. The only thing that then came to mind was this criterion which tells us the existence of the limit even without proving what the limit is.

Also, even though I allow $a$ and $b$ to take values in the extended real line, I am assuming that all the limits involved are finite.