Post History
#10: Post edited
by
bharathk98
·
2026-05-08T17:02:57Z (4 months ago)
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- **Edit 1:** The title does not match *Question 1* at the bottom of this post. To see an answer to the title, [see the following](https://math.codidact.com/posts/295434/295457#answer-295457). For clarfications on Question 1, see the following (**Edit 2**).
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- We want $\mathcal{G}$ to be similar to $f$ in the context, except $\mathcal{G}$ is non-Lebesgue integrable on any interval.
**Edit 2:** We wish to define an everywhere surjective function (without axiom of choice) whose Lebesgue measure is undefined on any interval- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- **Edit 1:** The title does not match *Question 1* at the bottom of this post. To see an answer to the title, [see the following](https://math.codidact.com/posts/295434/295457#answer-295457). For clarfications on Question 1, see the following (**Edit 2**).
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- We want $\mathcal{G}$ to be similar to $f$ in the context, except $\mathcal{G}$ is non-Lebesgue integrable on any interval.
- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#9: Post edited
by
bharathk98
·
2026-02-20T20:10:27Z (7 months ago)
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f\right|_{(c,d)}$ has an undefined expected value.
**Edit 1:** The title does not match *Question 1* at the bottom of this post. To see an answer to the title, [see the following](https://math.codidact.com/posts/295434/295457#answer-295457). For clarfications on Question 1, see the the following (**Edit 2**).- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- We want $\mathcal{G}$ to be similar to $f$ in the context, except $\mathcal{G}$ is non-Lebesgue integrable on any interval.
**Edit 2:** We want to define an everywhere surjective function (without axiom of choice) whose Lebesgue measure is undefined on any interval- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- **Edit 1:** The title does not match *Question 1* at the bottom of this post. To see an answer to the title, [see the following](https://math.codidact.com/posts/295434/295457#answer-295457). For clarfications on Question 1, see the following (**Edit 2**).
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- We want $\mathcal{G}$ to be similar to $f$ in the context, except $\mathcal{G}$ is non-Lebesgue integrable on any interval.
- **Edit 2:** We wish to define an everywhere surjective function (without axiom of choice) whose Lebesgue measure is undefined on any interval
- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#8: Post edited
by
bharathk98
·
2026-02-20T20:09:39Z (7 months ago)
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
Similar to $f$ in the context, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- **Edit 1:** The title does not match *Question 1* at the bottom of this post. To see an answer to the title, [see the following](https://math.codidact.com/posts/295434/295457#answer-295457). For clarfications on Question 1, see the the following (**Edit 2**).
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- We want $\mathcal{G}$ to be similar to $f$ in the context, except $\mathcal{G}$ is non-Lebesgue integrable on any interval.
- **Edit 2:** We want to define an everywhere surjective function (without axiom of choice) whose Lebesgue measure is undefined on any interval
- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#7: Post edited
by
bharathk98
·
2026-02-11T21:36:36Z (7 months ago)
**Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424) $\left.f ight|_{(c,d)}$ has an undefined expected value.- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Similar to $f$ in the context, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424/295428#answer-295428) $\left.f ight|_{(c,d)}$ has an undefined expected value.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Similar to $f$ in the context, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#6: Post edited
by
bharathk98
·
2026-02-11T21:21:34Z (7 months ago)
Defining a function that is not Lebesgue integrable on any interval?
- Defining a explicit function, without axiom of choice, that is not Lebesgue integrable on any interval?
**Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), [disproves](https://www.reddit.com/r/askmath/comments/1r18fbq/comment/o4o10qj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) the claim $\left.f ight|_{(c,d)}$ has an undefined mean.- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.**Question 1:** How do we define a function that satisfies the two properties,1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424) $\left.f ight|_{(c,d)}$ has an undefined expected value.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Similar to $f$ in the context, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define an explicit $\mathcal{G}:\mathbb{R}\to\mathbb{R}$ (without axiom of choice) that satisfies two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#5: Post edited
by
bharathk98
·
2026-02-11T21:00:09Z (7 months ago)
**Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), [disproves](https://www.reddit.com/r/askmath/comments/1r18fbq/comment/o4o10qj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) the claim $\left.f ight|_{(c,d)}$ has an undefined mean.- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/a/476609/87856) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), [disproves](https://www.reddit.com/r/askmath/comments/1r18fbq/comment/o4o10qj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) the claim $\left.f ight|_{(c,d)}$ has an undefined mean.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#4: Post edited
by
bharathk98
·
2026-02-11T19:35:26Z (7 months ago)
**Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), [disproves](https://www.reddit.com/r/askmath/comments/1r18fbq/comment/o4o10qj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) the claim $\left.f ight|_{(c,d)}$ has an undefined expected value.- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), [disproves](https://www.reddit.com/r/askmath/comments/1r18fbq/comment/o4o10qj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) the claim $\left.f ight|_{(c,d)}$ has an undefined mean.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#3: Post edited
by
bharathk98
·
2026-02-11T19:29:09Z (7 months ago)
**Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424) $\left.f ight|_{(c,d)}$ has an undefined expected value.- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), [disproves](https://www.reddit.com/r/askmath/comments/1r18fbq/comment/o4o10qj/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) the claim $\left.f ight|_{(c,d)}$ has an undefined expected value.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#2: Post edited
by
bharathk98
·
2026-02-11T18:15:00Z (7 months ago)
- **Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
Similar to $f$ in the context, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.- **Question 1:** How do we define a function that satisfies the two properties,
1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
- **Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424) $\left.f\right|_{(c,d)}$ has an undefined expected value.
- -----
- To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
- Then, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
- **Question 1:** How do we define a function that satisfies the two properties,
- 1. The restriction of $\mathcal{G}$ to any interval has an infinite area both above and below the $x$-axis.
- 2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?
#1: Initial revision
by
bharathk98
·
2026-02-11T18:10:04Z (7 months ago)
Defining a function that is not Lebesgue integrable on any interval?
**Context:** This [example](https://mathoverflow.net/questions/476471/is-there-an-explicit-everywhere-surjective-f-mathbbr-to-mathbbr-whose-gr) of an everywhere surjective $f:\mathbb{R}\to\mathbb{R}$, whose graph has zero Hausdorff measure in its dimension (i.e., the measure is defined on the Borel $\sigma$-algebra), disproves [the claim](https://math.codidact.com/posts/295424) $\left.f\right|_{(c,d)}$ has an undefined expected value.
-----
To change this incorrect assumption, replace $f$ with the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$. Let $\lambda(\cdot)$ be the Lebesgue measure defined on the Borel $\sigma$-algebra.
Similar to $f$ in the context, we want $\mathcal{G}$ to be non-Lebesgue integrable on any interval.
**Question 1:** How do we define a function that satisfies the two properties,
1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below the $x$-axis.
2. For all $a\lt b$ and $c \lt d$ real numbers, the set $\{x \in (a,b):\mathcal{G}(x) \in (c,d)\}$ has a positive Lebesgue measure?