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<jats:inline-formula><jats:alternatives><jats:tex-math>$$f_1,f_2,\\ldots :I \\rightarrow \\mathbb {R}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>R<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (<jats:italic>I<\/jats:italic> is a fixed closed interval) with <jats:inline-formula><jats:alternatives><jats:tex-math>$$f_1\\le f_2 \\le \\cdots $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mo>\u22ef<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> define a set <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} I_f:=\\big \\{x \\in I : \\lim _{n \\rightarrow \\infty } f_n(x)=+\\infty \\big \\}. \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtable>\n                      <mml:mtr>\n                        <mml:mtd>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>I<\/mml:mi>\n                              <mml:mi>f<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:munder>\n                              <mml:mo>lim<\/mml:mo>\n                              <mml:mrow>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\u2192<\/mml:mo>\n                                <mml:mi>\u221e<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:msub>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\u221e<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mtd>\n                      <\/mml:mtr>\n                    <\/mml:mtable>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:disp-formula>We study the properties of the family of all admissible <jats:inline-formula><jats:alternatives><jats:tex-math>$$I_f$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:mi>f<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-s and the family of all admissible <jats:inline-formula><jats:alternatives><jats:tex-math>$$I_f$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>I<\/mml:mi>\n                    <mml:mi>f<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-s under the additional assumption <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\lim _{n \\rightarrow \\infty } \\int _x^y f_n(t)\\,dt=+\\infty \\quad \\text { for all }x,y \\in I\\text { with }x&lt;y. \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtable>\n                      <mml:mtr>\n                        <mml:mtd>\n                          <mml:mrow>\n                            <mml:munder>\n                              <mml:mo>lim<\/mml:mo>\n                              <mml:mrow>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>\u2192<\/mml:mo>\n                                <mml:mi>\u221e<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munder>\n                            <mml:msubsup>\n                              <mml:mo>\u222b<\/mml:mo>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>y<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:msub>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mspace \/>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>\u221e<\/mml:mi>\n                            <mml:mspace \/>\n                            <mml:mspace \/>\n   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